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r""" Factory
This file contains the constructor classes and functions for `p`-adic rings and fields.
AUTHORS:
- David Roe
TESTS::
sage: R = ZpLC(2) doctest:...: FutureWarning: This class/method/function is marked as experimental. It, its functionality or its interface might change without a formal deprecation. See http://trac.sagemath.org/23505 for details. sage: R = ZpLF(2) sage: R = QpLC(2) sage: R = QpLF(2) """
#***************************************************************************** # Copyright (C) 2007-2013 David Roe <roed.math@gmail.com> # William Stein <wstein@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # # http://www.gnu.org/licenses/ #***************************************************************************** from __future__ import absolute_import, print_function
from sage.misc.superseded import experimental
from sage.structure.factory import UniqueFactory from sage.rings.integer import Integer from sage.rings.infinity import Infinity from sage.structure.factorization import Factorization from sage.rings.integer_ring import ZZ from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing from sage.rings.polynomial.polynomial_element import is_Polynomial from sage.structure.element import is_Element from .padic_base_leaves import (pAdicRingCappedRelative, pAdicRingCappedAbsolute, pAdicRingFixedMod, pAdicRingFloatingPoint, pAdicRingLattice, pAdicFieldCappedRelative, pAdicFieldFloatingPoint, pAdicFieldLattice) from . import padic_printing
###################################################### # ext_table -- # This dictionary controls what class is created by the extension # factory when it finds a given class in the ground ring of the tower. ######################################################
from .padic_extension_leaves import * from functools import reduce #This imports all of the classes used in the ext_table below.
ext_table = {} ext_table['e', pAdicFieldCappedRelative] = EisensteinExtensionFieldCappedRelative ext_table['e', pAdicRingCappedAbsolute] = EisensteinExtensionRingCappedAbsolute ext_table['e', pAdicRingCappedRelative] = EisensteinExtensionRingCappedRelative ext_table['e', pAdicRingFixedMod] = EisensteinExtensionRingFixedMod #ext_table['e', pAdicRingFloatingPoint] = EisensteinExtensionRingFloatingPoint #ext_table['e', pAdicFieldFloatingPoint] = EisensteinExtensionFieldFloatingPoint #ext_table['p', pAdicFieldCappedRelative] = pAdicGeneralExtensionFieldCappedRelative #ext_table['p', pAdicRingCappedAbsolute] = pAdicGeneralExtensionRingCappedAbsolute #ext_table['p', pAdicRingCappedRelative] = pAdicGeneralExtensionRingCappedRelative #ext_table['p', pAdicRingFixedMod] = pAdicGeneralExtensionRingFixedMod ext_table['u', pAdicFieldCappedRelative] = UnramifiedExtensionFieldCappedRelative ext_table['u', pAdicRingCappedAbsolute] = UnramifiedExtensionRingCappedAbsolute ext_table['u', pAdicRingCappedRelative] = UnramifiedExtensionRingCappedRelative ext_table['u', pAdicRingFixedMod] = UnramifiedExtensionRingFixedMod ext_table['u', pAdicRingFloatingPoint] = UnramifiedExtensionRingFloatingPoint ext_table['u', pAdicFieldFloatingPoint] = UnramifiedExtensionFieldFloatingPoint
def _default_show_prec(type, print_mode): r""" Returns the default show_prec value for a given type and printing mode.
INPUT:
- ``type`` -- a string: ``'capped-rel'``, ``'capped-abs'``, ``'fixed-mod'``, ``'floating-point'``, ``'lattice-cap'`` or ``'lattice-float'`` - ``print_mode`` -- a string: ``'series'``, ``'terse'``, ``'val-unit'``, ``'digits'``, ``'bars'``
EXAMPLES::
sage: from sage.rings.padics.factory import _default_show_prec sage: _default_show_prec('floating-point', 'series') False sage: _default_show_prec('capped-rel', 'series') True sage: _default_show_prec('capped-abs', 'digits') False """ else:
def get_key_base(p, prec, type, print_mode, names, ram_name, print_pos, print_sep, print_alphabet, print_max_terms, show_prec, check, valid_types, label=None): r""" This implements create_key for Zp and Qp: moving it here prevents code duplication.
It fills in unspecified values and checks for contradictions in the input. It also standardizes irrelevant options so that duplicate parents are not created.
EXAMPLES::
sage: from sage.rings.padics.factory import get_key_base sage: get_key_base(11, 5, 'capped-rel', None, None, None, None, ':', None, None, False, True, ['capped-rel']) (11, 5, 'capped-rel', 'series', '11', True, '|', (), -1, False, None) sage: get_key_base(12, 5, 'capped-rel', 'digits', None, None, None, None, None, None, True, False, ['capped-rel']) (12, 5, 'capped-rel', 'digits', '12', True, '|', ('0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B'), -1, True, None) """ except TypeError: raise TypeError("relative cap must be either a positive integer or infinity") raise ValueError("relative cap must be positive") except TypeError: raise TypeError("absolute cap must be an integer") relative_cap = Infinity else: prec = (DEFAULT_PREC, 2*DEFAULT_PREC) else: print_max_terms = print_mode['max_terms'] else: print_mode = None
# We eliminate irrelevant print options (e.g. print_pos if p = 2) else:
names = str(names) print_ram_name = str(print_ram_name) raise ValueError("If both names (%s) and print_ram_name (%s) are specified, they must agree"%(names, print_ram_name)) else: names = print_ram_name else: name = str(names) else: raise ValueError("type must be %s"%(", ".join(valid_types)))
####################################################################################################### # # p-Adic Fields # Qp -- base field # Qq -- unramified extension field of Qp # QpCR, QpLC, QpLF, QqCR -- shortcuts for capped relative and lattice versions of Qp and Qq # #######################################################################################################
padic_field_cache = {} DEFAULT_PREC = Integer(20) class Qp_class(UniqueFactory): r""" A creation function for `p`-adic fields.
INPUT:
- ``p`` -- integer: the `p` in `\mathbb{Q}_p`
- ``prec`` -- integer (default: ``20``) the precision cap of the field. In the lattice capped case, ``prec`` can either be a pair (``relative_cap``, ``absolute_cap``) or an integer (understood at relative cap). Except in the floating point case, individual elements keep track of their own precision. See TYPES and PRECISION below.
- ``type`` -- string (default: ``'capped-rel'``) Valid types are ``'capped-rel'``, ``'floating-point'``, ``'lattice-cap'``, ``'lattice-float'``. See TYPES and PRECISION below
- ``print_mode`` -- string (default: ``None``). Valid modes are 'series', 'val-unit', 'terse', 'digits', and 'bars'. See PRINTING below
- ``names`` -- string or tuple (defaults to a string representation of `p`). What to use whenever `p` is printed.
- ``ram_name`` -- string. Another way to specify the name; for consistency with the ``Qq`` and ``Zq`` and extension functions.
- ``print_pos`` -- bool (default ``None``) Whether to only use positive integers in the representations of elements. See PRINTING below.
- ``print_sep`` -- string (default ``None``) The separator character used in the ``'bars'`` mode. See PRINTING below.
- ``print_alphabet`` -- tuple (default ``None``) The encoding into digits for use in the 'digits' mode. See PRINTING below.
- ``print_max_terms`` -- integer (default ``None``) The maximum number of terms shown. See PRINTING below.
- ``show_prec`` -- bool (default ``None``) whether to show the precision for elements. See PRINTING below.
- ``check`` -- bool (default ``True``) whether to check if `p` is prime. Non-prime input may cause seg-faults (but can also be useful for base n expansions for example)
- ``label`` -- string (default ``None``) used for lattice precision to create parents with different lattices.
OUTPUT:
- The corresponding `p`-adic field.
TYPES AND PRECISION:
There are two types of precision for a `p`-adic element. The first is relative precision, which gives the number of known `p`-adic digits::
sage: R = Qp(5, 20, 'capped-rel', 'series'); a = R(675); a 2*5^2 + 5^4 + O(5^22) sage: a.precision_relative() 20
The second type of precision is absolute precision, which gives the power of `p` that this element is defined modulo::
sage: a.precision_absolute() 22
There are three types of `p`-adic fields: capped relative fields, floating point fields and lattice precision fields.
In the capped relative case, the relative precision of an element is restricted to be at most a certain value, specified at the creation of the field. Individual elements also store their own precision, so the effect of various arithmetic operations on precision is tracked. When you cast an exact element into a capped relative field, it truncates it to the precision cap of the field.::
sage: R = Qp(5, 5, 'capped-rel', 'series'); a = R(4006); a 1 + 5 + 2*5^3 + 5^4 + O(5^5) sage: b = R(4025); b 5^2 + 2*5^3 + 5^4 + 5^5 + O(5^7) sage: a + b 1 + 5 + 5^2 + 4*5^3 + 2*5^4 + O(5^5)
In the floating point case, elements do not track their precision, but the relative precision of elements is truncated during arithmetic to the precision cap of the field.
In the lattice case, precision on elements is tracked by a global lattice that is updated after every operation, yielding better precision behavior at the cost of higher memory and runtime usage.
PRINTING:
There are many different ways to print `p`-adic elements. The way elements of a given field print is controlled by options passed in at the creation of the field. There are five basic printing modes (series, val-unit, terse, digits and bars), as well as various options that either hide some information in the print representation or sometimes make print representations more compact. Note that the printing options affect whether different `p`-adic fields are considered equal.
1. **series**: elements are displayed as series in `p`.::
sage: R = Qp(5, print_mode='series'); a = R(70700); a 3*5^2 + 3*5^4 + 2*5^5 + 4*5^6 + O(5^22) sage: b = R(-70700); b 2*5^2 + 4*5^3 + 5^4 + 2*5^5 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + 4*5^11 + 4*5^12 + 4*5^13 + 4*5^14 + 4*5^15 + 4*5^16 + 4*5^17 + 4*5^18 + 4*5^19 + 4*5^20 + 4*5^21 + O(5^22)
*print_pos* controls whether negatives can be used in the coefficients of powers of `p`.::
sage: S = Qp(5, print_mode='series', print_pos=False); a = S(70700); a -2*5^2 + 5^3 - 2*5^4 - 2*5^5 + 5^7 + O(5^22) sage: b = S(-70700); b 2*5^2 - 5^3 + 2*5^4 + 2*5^5 - 5^7 + O(5^22)
*print_max_terms* limits the number of terms that appear.::
sage: T = Qp(5, print_mode='series', print_max_terms=4); b = R(-70700); repr(b) '2*5^2 + 4*5^3 + 5^4 + 2*5^5 + ... + O(5^22)'
*names* affects how the prime is printed.::
sage: U.<p> = Qp(5); p p + O(p^21)
*show_prec* determines whether the precision is printed (default ``True``)::
sage: Qp(5, show_prec=False)(6) 1 + 5
*print_sep* and *print_alphabet* have no effect in series mode.
Note that print options affect equality::
sage: R == S, R == T, R == U, S == T, S == U, T == U (False, False, False, False, False, False)
2. **val-unit**: elements are displayed as p^k*u::
sage: R = Qp(5, print_mode='val-unit'); a = R(70700); a 5^2 * 2828 + O(5^22) sage: b = R(-707/5); b 5^-1 * 95367431639918 + O(5^19)
*print_pos* controls whether to use a balanced representation or not.::
sage: S = Qp(5, print_mode='val-unit', print_pos=False); b = S(-70700); b 5^2 * (-2828) + O(5^22)
*names* affects how the prime is printed.::
sage: T = Qp(5, print_mode='val-unit', names='pi'); a = T(70700); a pi^2 * 2828 + O(pi^22)
*show_prec* determines whether the precision is printed (default ``True``)::
sage: Qp(5, print_mode='val-unit', show_prec=False)(30) 5 * 6
*print_max_terms*, *print_sep* and *print_alphabet* have no effect.
Equality again depends on the printing options::
sage: R == S, R == T, S == T (False, False, False)
3. **terse**: elements are displayed as an integer in base 10 or the quotient of an integer by a power of `p` (still in base 10)::
sage: R = Qp(5, print_mode='terse'); a = R(70700); a 70700 + O(5^22) sage: b = R(-70700); b 2384185790944925 + O(5^22) sage: c = R(-707/5); c 95367431639918/5 + O(5^19)
The denominator, as of version 3.3, is always printed explicitly as a power of `p`, for predictability.::
sage: d = R(707/5^2); d 707/5^2 + O(5^18)
*print_pos* controls whether to use a balanced representation or not.::
sage: S = Qp(5, print_mode='terse', print_pos=False); b = S(-70700); b -70700 + O(5^22) sage: c = S(-707/5); c -707/5 + O(5^19)
*name* affects how the name is printed.::
sage: T.<unif> = Qp(5, print_mode='terse'); c = T(-707/5); c 95367431639918/unif + O(unif^19) sage: d = T(-707/5^10); d 95367431639918/unif^10 + O(unif^10)
*show_prec* determines whether the precision is printed (default ``True``)::
sage: Qp(5, print_mode='terse', show_prec=False)(6) 6
*print_max_terms*, *print_sep* and *print_alphabet* have no effect.
Equality depends on printing options::
sage: R == S, R == T, S == T (False, False, False)
4. **digits**: elements are displayed as a string of base `p` digits
Restriction: you can only use the digits printing mode for small primes. Namely, `p` must be less than the length of the alphabet tuple (default alphabet has length 62).::
sage: R = Qp(5, print_mode='digits'); a = R(70700); repr(a) '...0000000000000004230300' sage: b = R(-70700); repr(b) '...4444444444444440214200' sage: c = R(-707/5); repr(c) '...4444444444444443413.3' sage: d = R(-707/5^2); repr(d) '...444444444444444341.33'
Observe that the significant 0's are printed even if they are located in front of the number. On the contrary, unknown digits located after the comma appears as question marks. The precision can therefore be read in this mode as well. Here are more examples::
sage: p = 7 sage: K = Qp(p, prec=10, print_mode='digits') sage: repr(K(1)) '...0000000001' sage: repr(K(p^2)) '...000000000100' sage: repr(K(p^-5)) '...00000.00001' sage: repr(K(p^-20)) '...?.??????????0000000001'
*print_max_terms* limits the number of digits that are printed. Note that if the valuation of the element is very negative, more digits will be printed.::
sage: S = Qp(5, print_mode='digits', print_max_terms=4); b = S(-70700); repr(b) '...214200' sage: d = S(-707/5^2); repr(d) '...41.33' sage: e = S(-707/5^6); repr(e) '...?.434133' sage: f = S(-707/5^6,absprec=-2); repr(f) '...?.??4133' sage: g = S(-707/5^4); repr(g) '...?.4133'
*print_alphabet* controls the symbols used to substitute for digits greater than 9.
Defaults to ('0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V', 'W', 'X', 'Y', 'Z', 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z')::
sage: T = Qp(5, print_mode='digits', print_max_terms=4, print_alphabet=('1','2','3','4','5')); b = T(-70700); repr(b) '...325311'
*show_prec* determines whether the precision is printed (default ``False``)::
sage: repr(Zp(5, print_mode='digits', show_prec=True)(6)) '...11 + O(5^20)'
*print_pos*, *name* and *print_sep* have no effect.
Equality depends on printing options::
sage: R == S, R == T, S == T (False, False, False)
5. **bars**: elements are displayed as a string of base `p` digits with separators::
sage: R = Qp(5, print_mode='bars'); a = R(70700); repr(a) '...4|2|3|0|3|0|0' sage: b = R(-70700); repr(b) '...4|4|4|4|4|4|4|4|4|4|4|4|4|4|4|0|2|1|4|2|0|0' sage: d = R(-707/5^2); repr(d) '...4|4|4|4|4|4|4|4|4|4|4|4|4|4|4|3|4|1|.|3|3'
Again, note that it's not possible to read of the precision from the representation in this mode.
*print_pos* controls whether the digits can be negative.::
sage: S = Qp(5, print_mode='bars',print_pos=False); b = S(-70700); repr(b) '...-1|0|2|2|-1|2|0|0'
*print_max_terms* limits the number of digits that are printed. Note that if the valuation of the element is very negative, more digits will be printed.::
sage: T = Qp(5, print_mode='bars', print_max_terms=4); b = T(-70700); repr(b) '...2|1|4|2|0|0' sage: d = T(-707/5^2); repr(d) '...4|1|.|3|3' sage: e = T(-707/5^6); repr(e) '...|.|4|3|4|1|3|3' sage: f = T(-707/5^6,absprec=-2); repr(f) '...|.|?|?|4|1|3|3' sage: g = T(-707/5^4); repr(g) '...|.|4|1|3|3'
*print_sep* controls the separation character.::
sage: U = Qp(5, print_mode='bars', print_sep=']['); a = U(70700); repr(a) '...4][2][3][0][3][0][0'
*show_prec* determines whether the precision is printed (default ``False``)::
sage: repr(Zp(5, print_mode='bars', show_prec=True)(6)) '...1|1 + O(5^20)'
*name* and *print_alphabet* have no effect.
Equality depends on printing options::
sage: R == S, R == T, R == U, S == T, S == U, T == U (False, False, False, False, False, False)
EXAMPLES::
sage: K = Qp(15, check=False); a = K(999); a 9 + 6*15 + 4*15^2 + O(15^20) """ def create_key(self, p, prec = None, type = 'capped-rel', print_mode = None, names = None, ram_name = None, print_pos = None, print_sep = None, print_alphabet = None, print_max_terms = None, show_prec = None, check = True, label = None): # specific to Lattice precision r""" Creates a key from input parameters for ``Qp``.
See the documentation for ``Qp`` for more information.
TESTS::
sage: Qp.create_key(5,40) (5, 40, 'capped-rel', 'series', '5', True, '|', (), -1, True, None) """ # old pickle; names is what used to be halt. names = ram_name ram_name = print_pos print_pos = print_sep print_alphabet = print_max_terms print_max_terms = check check = True raise ValueError("label keyword only supported for lattice precision")
def create_object(self, version, key): r""" Creates an object using a given key.
See the documentation for ``Qp`` for more information.
TESTS::
sage: Qp.create_object((3,4,2),(5, 41, 'capped-rel', 'series', '5', True, '|', (), -1)) 5-adic Field with capped relative precision 41 """ p, prec, type, print_mode, name = key print_pos, print_sep, print_alphabet, print_max_terms = None, None, None, None else: (len(version) > 2 and version[0] == 4 and version[1] == 5 and version[2] < 3)): # keys changed in order to reduce irrelevant duplications: e.g. two Qps with print_mode 'series' # that are identical except for different 'print_alphabet' now return the same object. print_max_terms, None, False, ['capped-rel', 'fixed-mod', 'capped-abs']) if obj is not None: return obj
'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name) else: 'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name) 'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name) else: 'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name) 'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name, label) else: 'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name, label) else: raise ValueError("unexpected type")
Qp = Qp_class("Qp")
###################################################### # Qq -- unramified extensions ######################################################
def Qq(q, prec = None, type = 'capped-rel', modulus = None, names=None, print_mode=None, ram_name = None, res_name = None, print_pos = None, print_sep = None, print_max_ram_terms = None, print_max_unram_terms = None, print_max_terse_terms = None, show_prec=None, check = True, implementation = 'FLINT'): r""" Given a prime power `q = p^n`, return the unique unramified extension of `\mathbb{Q}_p` of degree `n`.
INPUT:
- ``q`` -- integer, list, tuple or ``Factorization`` object. If ``q`` is an integer, it is the prime power `q` in `\mathbb{Q}_q`. If ``q`` is a ``Factorization`` object, it is the factorization of the prime power `q`. As a tuple it is the pair ``(p, n)``, and as a list it is a single element list ``[(p, n)]``.
- ``prec`` -- integer (default: ``20``) the precision cap of the field. Individual elements keep track of their own precision. See TYPES and PRECISION below.
- ``type`` -- string (default: ``'capped-rel'``) Valid types are ``'capped-rel'``, ``'floating-point'``, ``'lattice-cap'`` and ``'lattice-float'``. See TYPES and PRECISION below
- ``modulus`` -- polynomial (default ``None``) A polynomial defining an unramified extension of `\mathbb{Q}_p`. See MODULUS below.
- ``names`` -- string or tuple (``None`` is only allowed when `q=p`). The name of the generator, reducing to a generator of the residue field.
- ``print_mode`` -- string (default: ``None``). Valid modes are ``'series'``, ``'val-unit'``, ``'terse'``, and ``'bars'``. See PRINTING below.
- ``ram_name`` -- string (defaults to string representation of `p` if None). ``ram_name`` controls how the prime is printed. See PRINTING below.
- ``res_name`` -- string (defaults to ``None``, which corresponds to adding a ``'0'`` to the end of the name). Controls how elements of the reside field print.
- ``print_pos`` -- bool (default ``None``) Whether to only use positive integers in the representations of elements. See PRINTING below.
- ``print_sep`` -- string (default ``None``) The separator character used in the ``'bars'`` mode. See PRINTING below.
- ``print_max_ram_terms`` -- integer (default ``None``) The maximum number of powers of `p` shown. See PRINTING below.
- ``print_max_unram_terms`` -- integer (default ``None``) The maximum number of entries shown in a coefficient of `p`. See PRINTING below.
- ``print_max_terse_terms`` -- integer (default ``None``) The maximum number of terms in the polynomial representation of an element (using ``'terse'``). See PRINTING below.
- ``show_prec`` -- bool (default ``None``) whether to show the precision for elements. See PRINTING below.
- ``check`` -- bool (default ``True``) whether to check inputs.
OUTPUT:
- The corresponding unramified `p`-adic field.
TYPES AND PRECISION:
There are two types of precision for a `p`-adic element. The first is relative precision, which gives the number of known `p`-adic digits::
sage: R.<a> = Qq(25, 20, 'capped-rel', print_mode='series'); b = 25*a; b a*5^2 + O(5^22) sage: b.precision_relative() 20
The second type of precision is absolute precision, which gives the power of `p` that this element is defined modulo::
sage: b.precision_absolute() 22
There are two types of unramified `p`-adic fields: capped relative fields, floating point fields.
In the capped relative case, the relative precision of an element is restricted to be at most a certain value, specified at the creation of the field. Individual elements also store their own precision, so the effect of various arithmetic operations on precision is tracked. When you cast an exact element into a capped relative field, it truncates it to the precision cap of the field.::
sage: R.<a> = Qq(9, 5, 'capped-rel', print_mode='series'); b = (1+2*a)^4; b 2 + (2*a + 2)*3 + (2*a + 1)*3^2 + O(3^5) sage: c = R(3249); c 3^2 + 3^4 + 3^5 + 3^6 + O(3^7) sage: b + c 2 + (2*a + 2)*3 + (2*a + 2)*3^2 + 3^4 + O(3^5)
In the floating point case, elements do not track their precision, but the relative precision of elements is truncated during arithmetic to the precision cap of the field.
MODULUS:
The modulus needs to define an unramified extension of `\mathbb{Q}_p`: when it is reduced to a polynomial over `\mathbb{F}_p` it should be irreducible.
The modulus can be given in a number of forms.
1. A **polynomial**.
The base ring can be `\mathbb{Z}`, `\mathbb{Q}`, `\mathbb{Z}_p`, `\mathbb{Q}_p`, `\mathbb{F}_p`.::
sage: P.<x> = ZZ[] sage: R.<a> = Qq(27, modulus = x^3 + 2*x + 1); R.modulus() (1 + O(3^20))*x^3 + (O(3^20))*x^2 + (2 + O(3^20))*x + (1 + O(3^20)) sage: P.<x> = QQ[] sage: S.<a> = Qq(27, modulus = x^3 + 2*x + 1) sage: P.<x> = Zp(3)[] sage: T.<a> = Qq(27, modulus = x^3 + 2*x + 1) sage: P.<x> = Qp(3)[] sage: U.<a> = Qq(27, modulus = x^3 + 2*x + 1) sage: P.<x> = GF(3)[] sage: V.<a> = Qq(27, modulus = x^3 + 2*x + 1)
Which form the modulus is given in has no effect on the unramified extension produced::
sage: R == S, S == T, T == U, U == V (True, True, True, False)
unless the precision of the modulus differs. In the case of V, the modulus is only given to precision 1, so the resulting field has a precision cap of 1.::
sage: V.precision_cap() 1 sage: U.precision_cap() 20 sage: P.<x> = Qp(3)[] sage: modulus = x^3 + (2 + O(3^7))*x + (1 + O(3^10)) sage: modulus (1 + O(3^20))*x^3 + (2 + O(3^7))*x + (1 + O(3^10)) sage: W.<a> = Qq(27, modulus = modulus); W.precision_cap() 7
2. The modulus can also be given as a **symbolic expression**.::
sage: x = var('x') sage: X.<a> = Qq(27, modulus = x^3 + 2*x + 1); X.modulus() (1 + O(3^20))*x^3 + (O(3^20))*x^2 + (2 + O(3^20))*x + (1 + O(3^20)) sage: X == R True
By default, the polynomial chosen is the standard lift of the generator chosen for `\mathbb{F}_q`.::
sage: GF(125, 'a').modulus() x^3 + 3*x + 3 sage: Y.<a> = Qq(125); Y.modulus() (1 + O(5^20))*x^3 + (O(5^20))*x^2 + (3 + O(5^20))*x + (3 + O(5^20))
However, you can choose another polynomial if desired (as long as the reduction to `\mathbb{F}_p[x]` is irreducible).::
sage: P.<x> = ZZ[] sage: Z.<a> = Qq(125, modulus = x^3 + 3*x^2 + x + 1); Z.modulus() (1 + O(5^20))*x^3 + (3 + O(5^20))*x^2 + (1 + O(5^20))*x + (1 + O(5^20)) sage: Y == Z False
PRINTING:
There are many different ways to print `p`-adic elements. The way elements of a given field print is controlled by options passed in at the creation of the field. There are four basic printing modes (``'series'``, ``'val-unit'``, ``'terse'`` and ``'bars'``; ``'digits'`` is not available), as well as various options that either hide some information in the print representation or sometimes make print representations more compact. Note that the printing options affect whether different `p`-adic fields are considered equal.
1. **series**: elements are displayed as series in `p`.::
sage: R.<a> = Qq(9, 20, 'capped-rel', print_mode='series'); (1+2*a)^4 2 + (2*a + 2)*3 + (2*a + 1)*3^2 + O(3^20) sage: -3*(1+2*a)^4 3 + a*3^2 + 3^3 + (2*a + 2)*3^4 + (2*a + 2)*3^5 + (2*a + 2)*3^6 + (2*a + 2)*3^7 + (2*a + 2)*3^8 + (2*a + 2)*3^9 + (2*a + 2)*3^10 + (2*a + 2)*3^11 + (2*a + 2)*3^12 + (2*a + 2)*3^13 + (2*a + 2)*3^14 + (2*a + 2)*3^15 + (2*a + 2)*3^16 + (2*a + 2)*3^17 + (2*a + 2)*3^18 + (2*a + 2)*3^19 + (2*a + 2)*3^20 + O(3^21) sage: ~(3*a+18) (a + 2)*3^-1 + 1 + 2*3 + (a + 1)*3^2 + 3^3 + 2*3^4 + (a + 1)*3^5 + 3^6 + 2*3^7 + (a + 1)*3^8 + 3^9 + 2*3^10 + (a + 1)*3^11 + 3^12 + 2*3^13 + (a + 1)*3^14 + 3^15 + 2*3^16 + (a + 1)*3^17 + 3^18 + O(3^19)
*print_pos* controls whether negatives can be used in the coefficients of powers of `p`.::
sage: S.<b> = Qq(9, print_mode='series', print_pos=False); (1+2*b)^4 -1 - b*3 - 3^2 + (b + 1)*3^3 + O(3^20) sage: -3*(1+2*b)^4 3 + b*3^2 + 3^3 + (-b - 1)*3^4 + O(3^21)
*ram_name* controls how the prime is printed.::
sage: T.<d> = Qq(9, print_mode='series', ram_name='p'); 3*(1+2*d)^4 2*p + (2*d + 2)*p^2 + (2*d + 1)*p^3 + O(p^21)
*print_max_ram_terms* limits the number of powers of `p` that appear.::
sage: U.<e> = Qq(9, print_mode='series', print_max_ram_terms=4); repr(-3*(1+2*e)^4) '3 + e*3^2 + 3^3 + (2*e + 2)*3^4 + ... + O(3^21)'
*print_max_unram_terms* limits the number of terms that appear in a coefficient of a power of `p`.::
sage: V.<f> = Qq(128, prec = 8, print_mode='series'); repr((1+f)^9) '(f^3 + 1) + (f^5 + f^4 + f^3 + f^2)*2 + (f^6 + f^5 + f^4 + f + 1)*2^2 + (f^5 + f^4 + f^2 + f + 1)*2^3 + (f^6 + f^5 + f^4 + f^3 + f^2 + f + 1)*2^4 + (f^5 + f^4)*2^5 + (f^6 + f^5 + f^4 + f^3 + f + 1)*2^6 + (f + 1)*2^7 + O(2^8)' sage: V.<f> = Qq(128, prec = 8, print_mode='series', print_max_unram_terms = 3); repr((1+f)^9) '(f^3 + 1) + (f^5 + f^4 + ... + f^2)*2 + (f^6 + f^5 + ... + 1)*2^2 + (f^5 + f^4 + ... + 1)*2^3 + (f^6 + f^5 + ... + 1)*2^4 + (f^5 + f^4)*2^5 + (f^6 + f^5 + ... + 1)*2^6 + (f + 1)*2^7 + O(2^8)' sage: V.<f> = Qq(128, prec = 8, print_mode='series', print_max_unram_terms = 2); repr((1+f)^9) '(f^3 + 1) + (f^5 + ... + f^2)*2 + (f^6 + ... + 1)*2^2 + (f^5 + ... + 1)*2^3 + (f^6 + ... + 1)*2^4 + (f^5 + f^4)*2^5 + (f^6 + ... + 1)*2^6 + (f + 1)*2^7 + O(2^8)' sage: V.<f> = Qq(128, prec = 8, print_mode='series', print_max_unram_terms = 1); repr((1+f)^9) '(f^3 + ...) + (f^5 + ...)*2 + (f^6 + ...)*2^2 + (f^5 + ...)*2^3 + (f^6 + ...)*2^4 + (f^5 + ...)*2^5 + (f^6 + ...)*2^6 + (f + ...)*2^7 + O(2^8)' sage: V.<f> = Qq(128, prec = 8, print_mode='series', print_max_unram_terms = 0); repr((1+f)^9 - 1 - f^3) '(...)*2 + (...)*2^2 + (...)*2^3 + (...)*2^4 + (...)*2^5 + (...)*2^6 + (...)*2^7 + O(2^8)'
*show_prec* determines whether the precision is printed (default ``True``)::
sage: U.<e> = Qq(9, 2, show_prec=False); repr(-3*(1+2*e)^4) '3 + e*3^2'
*print_sep* and *print_max_terse_terms* have no effect.
Note that print options affect equality::
sage: R == S, R == T, R == U, R == V, S == T, S == U, S == V, T == U, T == V, U == V (False, False, False, False, False, False, False, False, False, False)
2. **val-unit**: elements are displayed as `p^k u`::
sage: R.<a> = Qq(9, 7, print_mode='val-unit'); b = (1+3*a)^9 - 1; b 3^3 * (15 + 64*a) + O(3^7) sage: ~b 3^-3 * (41 + a) + O(3)
*print_pos* controls whether to use a balanced representation or not.::
sage: S.<a> = Qq(9, 7, print_mode='val-unit', print_pos=False); b = (1+3*a)^9 - 1; b 3^3 * (15 - 17*a) + O(3^7) sage: ~b 3^-3 * (-40 + a) + O(3)
*ram_name* affects how the prime is printed.::
sage: A.<x> = Qp(next_prime(10^6), print_mode='val-unit')[] sage: T.<a> = Qq(next_prime(10^6)^3, 4, print_mode='val-unit', ram_name='p', modulus=x^3+385831*x^2+106556*x+321036); b = ~(next_prime(10^6)^2*(a^2 + a - 4)); b p^-2 * (503009563508519137754940 + 704413692798200940253892*a + 968097057817740999537581*a^2) + O(p^2) sage: b * (a^2 + a - 4) p^-2 * 1 + O(p^2)
*print_max_terse_terms* controls how many terms of the polynomial appear in the unit part.::
sage: U.<a> = Qq(17^4, 6, print_mode='val-unit', print_max_terse_terms=3); b = ~(17*(a^3-a+14)); b 17^-1 * (22110411 + 11317400*a + 20656972*a^2 + ...) + O(17^5) sage: b*17*(a^3-a+14) 1 + O(17^6)
*show_prec* determines whether the precision is printed (default ``True``)::
sage: U.<e> = Qq(9, 2, print_mode='val-unit', show_prec=False); repr(-3*(1+2*e)^4) '3 * (1 + 3*e)'
*print_sep*, *print_max_ram_terms* and *print_max_unram_terms* have no effect.
Equality again depends on the printing options::
sage: R == S, R == T, R == U, S == T, S == U, T == U (False, False, False, False, False, False)
3. **terse**: elements are displayed as a polynomial of degree less than the degree of the extension.::
sage: R.<a> = Qq(125, print_mode='terse') sage: (a+5)^177 68210977979428 + 90313850704069*a + 73948093055069*a^2 + O(5^20) sage: (a/5+1)^177 68210977979428/5^177 + 90313850704069/5^177*a + 73948093055069/5^177*a^2 + O(5^-157)
As of version 3.3, if coefficients of the polynomial are non-integral, they are always printed with an explicit power of `p` in the denominator.::
sage: 5*a + a^2/25 5*a + 1/5^2*a^2 + O(5^18)
*print_pos* controls whether to use a balanced representation or not.::
sage: (a-5)^6 22864 + 95367431627998*a + 8349*a^2 + O(5^20) sage: S.<a> = Qq(125, print_mode='terse', print_pos=False); b = (a-5)^6; b 22864 - 12627*a + 8349*a^2 + O(5^20) sage: (a - 1/5)^6 -20624/5^6 + 18369/5^5*a + 1353/5^3*a^2 + O(5^14)
*ram_name* affects how the prime is printed.::
sage: T.<a> = Qq(125, print_mode='terse', ram_name='p'); (a - 1/5)^6 95367431620001/p^6 + 18369/p^5*a + 1353/p^3*a^2 + O(p^14)
*print_max_terse_terms* controls how many terms of the polynomial are shown.::
sage: U.<a> = Qq(625, print_mode='terse', print_max_terse_terms=2); (a-1/5)^6 106251/5^6 + 49994/5^5*a + ... + O(5^14)
*show_prec* determines whether the precision is printed (default ``True``)::
sage: U.<e> = Qq(9, 2, print_mode='terse', show_prec=False); repr(-3*(1+2*e)^4) '3 + 9*e'
*print_sep*, *print_max_ram_terms* and *print_max_unram_terms* have no effect.
Equality again depends on the printing options::
sage: R == S, R == T, R == U, S == T, S == U, T == U (False, False, False, False, False, False)
4. **digits**: This print mode is not available when the residue field is not prime.
It might make sense to have a dictionary for small fields, but this isn't implemented.
5. **bars**: elements are displayed in a similar fashion to series, but more compactly.::
sage: R.<a> = Qq(125); (a+5)^6 (4*a^2 + 3*a + 4) + (3*a^2 + 2*a)*5 + (a^2 + a + 1)*5^2 + (3*a + 2)*5^3 + (3*a^2 + a + 3)*5^4 + (2*a^2 + 3*a + 2)*5^5 + O(5^20) sage: R.<a> = Qq(125, print_mode='bars', prec=8); repr((a+5)^6) '...[2, 3, 2]|[3, 1, 3]|[2, 3]|[1, 1, 1]|[0, 2, 3]|[4, 3, 4]' sage: repr((a-5)^6) '...[0, 4]|[1, 4]|[2, 0, 2]|[1, 4, 3]|[2, 3, 1]|[4, 4, 3]|[2, 4, 4]|[4, 3, 4]'
Note that elements with negative valuation are shown with a decimal point at valuation 0.::
sage: repr((a+1/5)^6) '...[3]|[4, 1, 3]|.|[1, 2, 3]|[3, 3]|[0, 0, 3]|[0, 1]|[0, 1]|[1]' sage: repr((a+1/5)^2) '...[0, 0, 1]|.|[0, 2]|[1]'
If not enough precision is known, ``'?'`` is used instead.::
sage: repr((a+R(1/5,relprec=3))^7) '...|.|?|?|?|?|[0, 1, 1]|[0, 2]|[1]'
Note that it's not possible to read of the precision from the representation in this mode.::
sage: b = a + 3; repr(b) '...[3, 1]' sage: c = a + R(3, 4); repr(c) '...[3, 1]' sage: b.precision_absolute() 8 sage: c.precision_absolute() 4
*print_pos* controls whether the digits can be negative.::
sage: S.<a> = Qq(125, print_mode='bars', print_pos=False); repr((a-5)^6) '...[1, -1, 1]|[2, 1, -2]|[2, 0, -2]|[-2, -1, 2]|[0, 0, -1]|[-2]|[-1, -2, -1]' sage: repr((a-1/5)^6) '...[0, 1, 2]|[-1, 1, 1]|.|[-2, -1, -1]|[2, 2, 1]|[0, 0, -2]|[0, -1]|[0, -1]|[1]'
*print_max_ram_terms* controls the maximum number of "digits" shown. Note that this puts a cap on the relative precision, not the absolute precision.::
sage: T.<a> = Qq(125, print_mode='bars', print_max_ram_terms=3, print_pos=False); repr((a-5)^6) '...[0, 0, -1]|[-2]|[-1, -2, -1]' sage: repr(5*(a-5)^6+50) '...[0, 0, -1]|[]|[-1, -2, -1]|[]'
However, if the element has negative valuation, digits are shown up to the decimal point.::
sage: repr((a-1/5)^6) '...|.|[-2, -1, -1]|[2, 2, 1]|[0, 0, -2]|[0, -1]|[0, -1]|[1]'
*print_sep* controls the separating character (``'|'`` by default).::
sage: U.<a> = Qq(625, print_mode='bars', print_sep=''); b = (a+5)^6; repr(b) '...[0, 1][4, 0, 2][3, 2, 2, 3][4, 2, 2, 4][0, 3][1, 1, 3][3, 1, 4, 1]'
*print_max_unram_terms* controls how many terms are shown in each "digit"::
sage: with local_print_mode(U, {'max_unram_terms': 3}): repr(b) '...[0, 1][4,..., 0, 2][3,..., 2, 3][4,..., 2, 4][0, 3][1,..., 1, 3][3,..., 4, 1]' sage: with local_print_mode(U, {'max_unram_terms': 2}): repr(b) '...[0, 1][4,..., 2][3,..., 3][4,..., 4][0, 3][1,..., 3][3,..., 1]' sage: with local_print_mode(U, {'max_unram_terms': 1}): repr(b) '...[..., 1][..., 2][..., 3][..., 4][..., 3][..., 3][..., 1]' sage: with local_print_mode(U, {'max_unram_terms':0}): repr(b-75*a) '...[...][...][...][...][][...][...]'
*show_prec* determines whether the precision is printed (default ``False``)::
sage: U.<e> = Qq(9, 2, print_mode='bars', show_prec=True); repr(-3*(1+2*e)^4) '...[0, 1]|[1]|[] + O(3^3)'
*ram_name* and *print_max_terse_terms* have no effect.
Equality depends on printing options::
sage: R == S, R == T, R == U, S == T, S == U, T == U (False, False, False, False, False, False)
EXAMPLES
Unlike for ``Qp``, you can't create ``Qq(N)`` when ``N`` is not a prime power.
However, you can use ``check=False`` to pass in a pair in order to not have to factor. If you do so, you need to use names explicitly rather than the ``R.<a>`` syntax.::
sage: p = next_prime(2^123) sage: k = Qp(p) sage: R.<x> = k[] sage: K = Qq([(p, 5)], modulus=x^5+x+4, names='a', ram_name='p', print_pos=False, check=False) sage: K.0^5 (-a - 4) + O(p^20)
In tests on ``sage.math.washington.edu``, the creation of ``K`` as above took an average of 1.58ms, while::
sage: K = Qq(p^5, modulus=x^5+x+4, names='a', ram_name='p', print_pos=False, check=True)
took an average of 24.5ms. Of course, with smaller primes these savings disappear.
TESTS:
Check that :trac:`8162` is resolved::
sage: R = Qq([(5,3)], names="alpha", check=False); R Unramified Extension in alpha defined by x^3 + 3*x + 3 with capped relative precision 20 over 5-adic Field sage: Qq((5, 3), names="alpha") is R True sage: Qq(125.factor(), names="alpha") is R True
Check that :trac:`18606` is resolved::
sage: x = QQ['x'].gen() sage: F = Qp(5,20) sage: K0 = F.extension(x^2-F(13),names = 'g') sage: K1 = F.extension(x^2-13,names = 'g') sage: K0 is K1 True """ raise ValueError("q must be a prime power") raise ValueError("q must be a factorization of a prime power") raise TypeError("q must be an integer, list, tuple or Factorization") raise ValueError("q must have shape [(p,k)]") raise ValueError("q must have shape (p,k)") q = tuple(q)
raise ValueError("q must be a prime power")
prec = Integer(prec)
print_sep=print_sep, print_max_terms=print_max_ram_terms, show_prec=show_prec, check=check)
return base
raise ValueError("must provide exactly one generator name") raise TypeError("You must specify the name of the generator.") names = str(names)
names=names, res_name=res_name, ram_name=ram_name, print_pos=print_pos, print_sep=print_sep, print_max_ram_terms=print_max_ram_terms, print_max_unram_terms=print_max_unram_terms, print_max_terse_terms=print_max_terse_terms, show_prec=show_prec, check=check, unram=True, implementation=implementation)
###################################################### # Short constructor names for different types ######################################################
def QpCR(p, prec = None, *args, **kwds): r""" A shortcut function to create capped relative `p`-adic fields.
Same functionality as ``Qp``. See documentation for ``Qp`` for a description of the input parameters.
EXAMPLES::
sage: QpCR(5, 40) 5-adic Field with capped relative precision 40 """
def QpFP(p, prec = None, *args, **kwds): r""" A shortcut function to create floating point `p`-adic fields.
Same functionality as ``Qp``. See documentation for ``Qp`` for a description of the input parameters.
EXAMPLES::
sage: QpFP(5, 40) 5-adic Field with floating precision 40 """
def QqCR(q, prec = None, *args, **kwds): r""" A shortcut function to create capped relative unramified `p`-adic fields.
Same functionality as ``Qq``. See documentation for ``Qq`` for a description of the input parameters.
EXAMPLES::
sage: R.<a> = QqCR(25, 40); R Unramified Extension in a defined by x^2 + 4*x + 2 with capped relative precision 40 over 5-adic Field """
def QqFP(q, prec = None, *args, **kwds): r""" A shortcut function to create floating point unramified `p`-adic fields.
Same functionality as ``Qq``. See documentation for ``Qq`` for a description of the input parameters.
EXAMPLES::
sage: R.<a> = QqFP(25, 40); R Unramified Extension in a defined by x^2 + 4*x + 2 with floating precision 40 over 5-adic Field """
@experimental(23505) def QpLC(p, prec = None, *args, **kwds): r""" A shortcut function to create `p`-adic fields with lattice precision.
See :func:`ZpLC` for more information about this model of precision.
EXAMPLES::
sage: R = QpLC(2) sage: R 2-adic Field with lattice-cap precision
"""
@experimental(23505) def QpLF(p, prec = None, *args, **kwds): r""" A shortcut function to create `p`-adic fields with lattice precision.
See :func:`ZpLC` for more information about this model of precision.
EXAMPLES::
sage: R = QpLF(2) sage: R 2-adic Field with lattice-float precision """
####################################################################################################### # # p-Adic Rings # Zp -- base rings # Zq -- unramified extension ring of Zp # ZpCR, ZpCA, ZpFM, ZpL, ZqCR, ZqCA, ZqFM, ZqL -- shortcuts for precision-type versions of Zp and Zq # #######################################################################################################
class Zp_class(UniqueFactory): r""" A creation function for `p`-adic rings.
INPUT:
- ``p`` -- integer: the `p` in `\mathbb{Z}_p`
- ``prec`` -- integer (default: ``20``) the precision cap of the ring. In the lattice capped case, ``prec`` can either be a pair (``relative_cap``, ``absolute_cap``) or an integer (understood at relative cap). Except for the fixed modulus and floating point cases, individual elements keep track of their own precision. See TYPES and PRECISION below.
- ``type`` -- string (default: ``'capped-rel'``) Valid types are ``'capped-rel'``, ``'capped-abs'``, ``'fixed-mod'``, ``'floating-point'``, ``'lattice-cap'``, ``'lattice-float'`` See TYPES and PRECISION below
- ``print_mode`` -- string (default: ``None``). Valid modes are ``'series'``, ``'val-unit'``, ``'terse'``, ``'digits'``, and ``'bars'``. See PRINTING below
- ``names`` -- string or tuple (defaults to a string representation of `p`). What to use whenever `p` is printed.
- ``print_pos`` -- bool (default ``None``) Whether to only use positive integers in the representations of elements. See PRINTING below.
- ``print_sep`` -- string (default ``None``) The separator character used in the ``'bars'`` mode. See PRINTING below.
- ``print_alphabet`` -- tuple (default ``None``) The encoding into digits for use in the ``'digits'`` mode. See PRINTING below.
- ``print_max_terms`` -- integer (default ``None``) The maximum number of terms shown. See PRINTING below.
- ``show_prec`` -- bool (default ``None``) whether to show the precision for elements. See PRINTING below.
- ``check`` -- bool (default ``True``) whether to check if `p` is prime. Non-prime input may cause seg-faults (but can also be useful for base `n` expansions for example)
- ``label`` -- string (default ``None``) used for lattice precision to create parents with different lattices.
OUTPUT:
- The corresponding `p`-adic ring.
TYPES AND PRECISION:
There are three types of precision. The first is relative precision; it gives the number of known `p`-adic digits::
sage: R = Zp(5, 20, 'capped-rel', 'series'); a = R(675); a 2*5^2 + 5^4 + O(5^22) sage: a.precision_relative() 20
The second type of precision is absolute precision, which gives the power of `p` that this element is defined modulo::
sage: a.precision_absolute() 22
The third one is lattice precision. It is not attached to a single `p`-adic number but is a unique object modeling the precision on a set of `p`-adics, which is typically the set of all elements within the same parent::
sage: R = ZpLC(17) sage: x = R(1,10); y = R(1,5) sage: R.precision() Precision lattice on 2 objects sage: R.precision().precision_lattice() [2015993900449 0] [ 0 1419857]
We refer to the documentation of the function :func:`ZpLC` for more information about this precision model.
There are many types of `p`-adic rings: capped relative rings (``type='capped-rel'``), capped absolute rings (``type='capped-abs'``), fixed modulus rings (``type='fixed-mod'``), floating point rings (``type='floating-point'``), lattice capped rings (``type='lattice-cap'``) and lattice float rings (``type='lattice-float'``).
In the capped relative case, the relative precision of an element is restricted to be at most a certain value, specified at the creation of the field. Individual elements also store their own precision, so the effect of various arithmetic operations on precision is tracked. When you cast an exact element into a capped relative field, it truncates it to the precision cap of the field.::
sage: R = Zp(5, 5, 'capped-rel', 'series'); a = R(4006); a 1 + 5 + 2*5^3 + 5^4 + O(5^5) sage: b = R(4025); b 5^2 + 2*5^3 + 5^4 + 5^5 + O(5^7) sage: a + b 1 + 5 + 5^2 + 4*5^3 + 2*5^4 + O(5^5)
In the capped absolute type, instead of having a cap on the relative precision of an element there is instead a cap on the absolute precision. Elements still store their own precisions, and as with the capped relative case, exact elements are truncated when cast into the ring.::
sage: R = Zp(5, 5, 'capped-abs', 'series'); a = R(4005); a 5 + 2*5^3 + 5^4 + O(5^5) sage: b = R(4025); b 5^2 + 2*5^3 + 5^4 + O(5^5) sage: a * b 5^3 + 2*5^4 + O(5^5) sage: (a * b) // 5^3 1 + 2*5 + O(5^2)
The fixed modulus type is the leanest of the `p`-adic rings: it is basically just a wrapper around `\mathbb{Z} / p^n \mathbb{Z}` providing a unified interface with the rest of the `p`-adics. This is the type you should use if your sole interest is speed. It does not track precision of elements.::
sage: R = Zp(5,5,'fixed-mod','series'); a = R(4005); a 5 + 2*5^3 + 5^4 + O(5^5) sage: a // 5 1 + 2*5^2 + 5^3 + O(5^5)
The floating point case is similar to the fixed modulus type in that elements do not trac their own precision. However, relative precision is truncated with each operation rather than absolute precision.
On the contrary, the lattice type tracks precision using lattices and automatic differentiation. It is rather slow but provides sharp (often optimal) results regarding precision. We refer to the documentation of the function :func:`ZpLC` for a small demonstration of the capabilities of this precision model.
PRINTING:
There are many different ways to print `p`-adic elements. The way elements of a given ring print is controlled by options passed in at the creation of the ring. There are five basic printing modes (series, val-unit, terse, digits and bars), as well as various options that either hide some information in the print representation or sometimes make print representations more compact. Note that the printing options affect whether different `p`-adic fields are considered equal.
1. **series**: elements are displayed as series in `p`.::
sage: R = Zp(5, print_mode='series'); a = R(70700); a 3*5^2 + 3*5^4 + 2*5^5 + 4*5^6 + O(5^22) sage: b = R(-70700); b 2*5^2 + 4*5^3 + 5^4 + 2*5^5 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + 4*5^11 + 4*5^12 + 4*5^13 + 4*5^14 + 4*5^15 + 4*5^16 + 4*5^17 + 4*5^18 + 4*5^19 + 4*5^20 + 4*5^21 + O(5^22)
*print_pos* controls whether negatives can be used in the coefficients of powers of `p`.::
sage: S = Zp(5, print_mode='series', print_pos=False); a = S(70700); a -2*5^2 + 5^3 - 2*5^4 - 2*5^5 + 5^7 + O(5^22) sage: b = S(-70700); b 2*5^2 - 5^3 + 2*5^4 + 2*5^5 - 5^7 + O(5^22)
*print_max_terms* limits the number of terms that appear.::
sage: T = Zp(5, print_mode='series', print_max_terms=4); b = R(-70700); b 2*5^2 + 4*5^3 + 5^4 + 2*5^5 + ... + O(5^22)
*names* affects how the prime is printed.::
sage: U.<p> = Zp(5); p p + O(p^21)
*show_prec* determines whether the precision is printed (default ``True``)::
sage: Zp(5, show_prec=False)(6) 1 + 5
*print_sep* and *print_alphabet* have no effect.
Note that print options affect equality::
sage: R == S, R == T, R == U, S == T, S == U, T == U (False, False, False, False, False, False)
2. **val-unit**: elements are displayed as `p^k u`::
sage: R = Zp(5, print_mode='val-unit'); a = R(70700); a 5^2 * 2828 + O(5^22) sage: b = R(-707*5); b 5 * 95367431639918 + O(5^21)
*print_pos* controls whether to use a balanced representation or not.::
sage: S = Zp(5, print_mode='val-unit', print_pos=False); b = S(-70700); b 5^2 * (-2828) + O(5^22)
*names* affects how the prime is printed.::
sage: T = Zp(5, print_mode='val-unit', names='pi'); a = T(70700); a pi^2 * 2828 + O(pi^22)
*show_prec* determines whether the precision is printed (default ``True``)::
sage: Zp(5, print_mode='val-unit', show_prec=False)(30) 5 * 6
*print_max_terms*, *print_sep* and *print_alphabet* have no effect.
Equality again depends on the printing options::
sage: R == S, R == T, S == T (False, False, False)
3. **terse**: elements are displayed as an integer in base 10::
sage: R = Zp(5, print_mode='terse'); a = R(70700); a 70700 + O(5^22) sage: b = R(-70700); b 2384185790944925 + O(5^22)
*print_pos* controls whether to use a balanced representation or not.::
sage: S = Zp(5, print_mode='terse', print_pos=False); b = S(-70700); b -70700 + O(5^22)
*name* affects how the name is printed. Note that this interacts with the choice of shorter string for denominators.::
sage: T.<unif> = Zp(5, print_mode='terse'); c = T(-707); c 95367431639918 + O(unif^20)
*show_prec* determines whether the precision is printed (default ``True``)::
sage: Zp(5, print_mode='terse', show_prec=False)(30) 30
*print_max_terms*, *print_sep* and *print_alphabet* have no effect.
Equality depends on printing options::
sage: R == S, R == T, S == T (False, False, False)
4. **digits**: elements are displayed as a string of base `p` digits
Restriction: you can only use the digits printing mode for small primes. Namely, `p` must be less than the length of the alphabet tuple (default alphabet has length 62).::
sage: R = Zp(5, print_mode='digits'); a = R(70700); repr(a) '...4230300' sage: b = R(-70700); repr(b) '...4444444444444440214200'
Note that it's not possible to read off the precision from the representation in this mode.
*print_max_terms* limits the number of digits that are printed.::
sage: S = Zp(5, print_mode='digits', print_max_terms=4); b = S(-70700); repr(b) '...214200'
*print_alphabet* controls the symbols used to substitute for digits greater than 9. Defaults to ('0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V', 'W', 'X', 'Y', 'Z', 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z')::
sage: T = Zp(5, print_mode='digits', print_max_terms=4, print_alphabet=('1','2','3','4','5')); b = T(-70700); repr(b) '...325311'
*show_prec* determines whether the precision is printed (default ``False``)::
sage: repr(Zp(5, 2, print_mode='digits', show_prec=True)(6)) '...11 + O(5^2)'
*print_pos*, *name* and *print_sep* have no effect.
Equality depends on printing options::
sage: R == S, R == T, S == T (False, False, False)
5. **bars**: elements are displayed as a string of base `p` digits with separators
sage: R = Zp(5, print_mode='bars'); a = R(70700); repr(a) '...4|2|3|0|3|0|0' sage: b = R(-70700); repr(b) '...4|4|4|4|4|4|4|4|4|4|4|4|4|4|4|0|2|1|4|2|0|0'
Again, note that it's not possible to read of the precision from the representation in this mode.
*print_pos* controls whether the digits can be negative.::
sage: S = Zp(5, print_mode='bars',print_pos=False); b = S(-70700); repr(b) '...-1|0|2|2|-1|2|0|0'
*print_max_terms* limits the number of digits that are printed.::
sage: T = Zp(5, print_mode='bars', print_max_terms=4); b = T(-70700); repr(b) '...2|1|4|2|0|0'
*print_sep* controls the separation character.::
sage: U = Zp(5, print_mode='bars', print_sep=']['); a = U(70700); repr(a) '...4][2][3][0][3][0][0'
*show_prec* determines whether the precision is printed (default ``False``)::
sage: repr(Zp(5, 2, print_mode='bars', show_prec=True)(6)) '1|1 + O(5^2)'
*name* and *print_alphabet* have no effect.
Equality depends on printing options::
sage: R == S, R == T, R == U, S == T, S == U, T == U (False, False, False, False, False, False)
EXAMPLES:
We allow non-prime `p`, but only if ``check = False``. Note that some features will not work.::
sage: K = Zp(15, check=False); a = K(999); a 9 + 6*15 + 4*15^2 + O(15^20)
We create rings with various parameters::
sage: Zp(7) 7-adic Ring with capped relative precision 20 sage: Zp(9) Traceback (most recent call last): ... ValueError: p must be prime sage: Zp(17, 5) 17-adic Ring with capped relative precision 5 sage: Zp(17, 5)(-1) 16 + 16*17 + 16*17^2 + 16*17^3 + 16*17^4 + O(17^5)
It works even with a fairly huge cap::
sage: Zp(next_prime(10^50), 100000) 100000000000000000000000000000000000000000000000151-adic Ring with capped relative precision 100000
We create each type of ring::
sage: Zp(7, 20, 'capped-rel') 7-adic Ring with capped relative precision 20 sage: Zp(7, 20, 'fixed-mod') 7-adic Ring of fixed modulus 7^20 sage: Zp(7, 20, 'capped-abs') 7-adic Ring with capped absolute precision 20
We create a capped relative ring with each print mode::
sage: k = Zp(7, 8, print_mode='series'); k 7-adic Ring with capped relative precision 8 sage: k(7*(19)) 5*7 + 2*7^2 + O(7^9) sage: k(7*(-19)) 2*7 + 4*7^2 + 6*7^3 + 6*7^4 + 6*7^5 + 6*7^6 + 6*7^7 + 6*7^8 + O(7^9)
::
sage: k = Zp(7, print_mode='val-unit'); k 7-adic Ring with capped relative precision 20 sage: k(7*(19)) 7 * 19 + O(7^21) sage: k(7*(-19)) 7 * 79792266297611982 + O(7^21)
::
sage: k = Zp(7, print_mode='terse'); k 7-adic Ring with capped relative precision 20 sage: k(7*(19)) 133 + O(7^21) sage: k(7*(-19)) 558545864083283874 + O(7^21)
Note that `p`-adic rings are cached (via weak references)::
sage: a = Zp(7); b = Zp(7) sage: a is b True
We create some elements in various rings::
sage: R = Zp(5); a = R(4); a 4 + O(5^20) sage: S = Zp(5, 10, type = 'capped-abs'); b = S(2); b 2 + O(5^10) sage: a + b 1 + 5 + O(5^10) """ def create_key(self, p, prec = None, type = 'capped-rel', print_mode = None, names = None, ram_name = None, print_pos = None, print_sep = None, print_alphabet = None, print_max_terms = None, show_prec = None, check = True, label = None): r""" Creates a key from input parameters for ``Zp``.
See the documentation for ``Zp`` for more information.
TESTS::
sage: Zp.create_key(5,40) (5, 40, 'capped-rel', 'series', '5', True, '|', (), -1, True, None) sage: Zp.create_key(5,40,print_mode='digits') (5, 40, 'capped-rel', 'digits', '5', True, '|', ('0', '1', '2', '3', '4'), -1, False, None) """ # old pickle; names is what used to be halt. names = ram_name ram_name = print_pos print_pos = print_sep print_alphabet = print_max_terms print_max_terms = check check = True raise ValueError("label keyword only supported for lattice precision") print_max_terms, show_prec, check, ['capped-rel', 'fixed-mod', 'capped-abs', 'floating-point', 'lattice-cap', 'lattice-float'], label=label)
def create_object(self, version, key): r""" Creates an object using a given key.
See the documentation for ``Zp`` for more information.
TESTS::
sage: Zp.create_object((3,4,2),(5, 41, 'capped-rel', 'series', '5', True, '|', (), -1)) 5-adic Ring with capped relative precision 41 """ (len(version) > 2 and version[0] == 3 and version[1] == 2 and version[2] < 3)): p, prec, type, print_mode, name = key print_pos, print_sep, print_alphabet, print_max_terms = None, None, None, None else: (len(version) > 2 and version[0] == 4 and version[1] == 5 and version[2] < 3)): # keys changed in order to reduce irrelevant duplications: e.g. two Zps with print_mode 'series' # that are identical except for different 'print_alphabet' now return the same object. print_max_terms, None, False, ['capped-rel', 'fixed-mod', 'capped-abs', 'lattice-cap', 'lattice-float']) if obj is not None: return obj 'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name) 'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name) 'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name) 'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name) 'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name, label) else: raise ValueError("unexpected type")
Zp = Zp_class("Zp")
###################################################### # Zq -- unramified extensions ######################################################
def Zq(q, prec = None, type = 'capped-rel', modulus = None, names=None, print_mode=None, ram_name = None, res_name = None, print_pos = None, print_sep = None, print_max_ram_terms = None, print_max_unram_terms = None, print_max_terse_terms = None, show_prec = None, check = True, implementation = 'FLINT'): r""" Given a prime power `q = p^n`, return the unique unramified extension of `\mathbb{Z}_p` of degree `n`.
INPUT:
- ``q`` -- integer, list or tuple: the prime power in `\mathbb{Q}_q`. Or a factorization object, single element list ``[(p, n)]`` where ``p`` is a prime and ``n`` a positive integer, or the pair ``(p, n)``.
- ``prec`` -- integer (default: ``20``) the precision cap of the field. Individual elements keep track of their own precision. See TYPES and PRECISION below.
- ``type`` -- string (default: ``'capped-rel'``) Valid types are ``'capped-abs'``, ``'capped-rel'``, ``'fixed-mod'``, and ``'floating-point'``. See TYPES and PRECISION below
- modulus -- polynomial (default None) A polynomial defining an unramified extension of `\mathbb{Z}_p`. See MODULUS below.
- ``names`` -- string or tuple (``None`` is only allowed when `q=p`). The name of the generator, reducing to a generator of the residue field.
- ``print_mode`` -- string (default: ``None``). Valid modes are ``'series'``, ``'val-unit'``, ``'terse'``, and ``'bars'``. See PRINTING below.
- ``ram_name`` -- string (defaults to string representation of `p` if None). ``ram_name`` controls how the prime is printed. See PRINTING below.
- ``res_name`` -- string (defaults to ``None``, which corresponds to adding a ``'0'`` to the end of the name). Controls how elements of the reside field print.
- ``print_pos`` -- bool (default ``None``) Whether to only use positive integers in the representations of elements. See PRINTING below.
- ``print_sep`` -- string (default ``None``) The separator character used in the ``'bars'`` mode. See PRINTING below.
- ``print_max_ram_terms`` -- integer (default ``None``) The maximum number of powers of `p` shown. See PRINTING below.
- ``print_max_unram_terms`` -- integer (default ``None``) The maximum number of entries shown in a coefficient of `p`. See PRINTING below.
- ``print_max_terse_terms`` -- integer (default ``None``) The maximum number of terms in the polynomial representation of an element (using ``'terse'``). See PRINTING below.
- ``show_prec`` -- bool (default ``None``) Whether to show the precision for elements. See PRINTING below.
- ``check`` -- bool (default ``True``) whether to check inputs.
- ``implementation`` -- string (default ``FLINT``) which implementation to use. ``NTL`` is the other option.
OUTPUT:
- The corresponding unramified `p`-adic ring.
TYPES AND PRECISION:
There are two types of precision for a `p`-adic element. The first is relative precision (default), which gives the number of known `p`-adic digits::
sage: R.<a> = Zq(25, 20, 'capped-rel', print_mode='series'); b = 25*a; b a*5^2 + O(5^22) sage: b.precision_relative() 20
The second type of precision is absolute precision, which gives the power of `p` that this element is defined modulo::
sage: b.precision_absolute() 22
There are many types of `p`-adic rings: capped relative rings (``type='capped-rel'``), capped absolute rings (``type='capped-abs'``), fixed modulus rings (``type='fixed-mod'``), and floating point rings (``type='floating-point'``).
In the capped relative case, the relative precision of an element is restricted to be at most a certain value, specified at the creation of the field. Individual elements also store their own precision, so the effect of various arithmetic operations on precision is tracked. When you cast an exact element into a capped relative field, it truncates it to the precision cap of the field.::
sage: R.<a> = Zq(9, 5, 'capped-rel', print_mode='series'); b = (1+2*a)^4; b 2 + (2*a + 2)*3 + (2*a + 1)*3^2 + O(3^5) sage: c = R(3249); c 3^2 + 3^4 + 3^5 + 3^6 + O(3^7) sage: b + c 2 + (2*a + 2)*3 + (2*a + 2)*3^2 + 3^4 + O(3^5)
One can invert non-units: the result is in the fraction field.::
sage: d = ~(3*b+c); d 2*3^-1 + (a + 1) + (a + 1)*3 + a*3^3 + O(3^4) sage: d.parent() Unramified Extension in a defined by x^2 + 2*x + 2 with capped relative precision 5 over 3-adic Field
The capped absolute case is the same as the capped relative case, except that the cap is on the absolute precision rather than the relative precision.::
sage: R.<a> = Zq(9, 5, 'capped-abs', print_mode='series'); b = 3*(1+2*a)^4; b 2*3 + (2*a + 2)*3^2 + (2*a + 1)*3^3 + O(3^5) sage: c = R(3249); c 3^2 + 3^4 + O(3^5) sage: b*c 2*3^3 + (2*a + 2)*3^4 + O(3^5) sage: b*c >> 1 2*3^2 + (2*a + 2)*3^3 + O(3^4)
The fixed modulus case is like the capped absolute, except that individual elements don't track their precision.::
sage: R.<a> = Zq(9, 5, 'fixed-mod', print_mode='series'); b = 3*(1+2*a)^4; b 2*3 + (2*a + 2)*3^2 + (2*a + 1)*3^3 + O(3^5) sage: c = R(3249); c 3^2 + 3^4 + O(3^5) sage: b*c 2*3^3 + (2*a + 2)*3^4 + O(3^5) sage: b*c >> 1 2*3^2 + (2*a + 2)*3^3 + O(3^5)
The floating point case is similar to the fixed modulus type in that elements do not trac their own precision. However, relative precision is truncated with each operation rather than absolute precision.
MODULUS:
The modulus needs to define an unramified extension of `\mathbb{Z}_p`: when it is reduced to a polynomial over `\mathbb{F}_p` it should be irreducible.
The modulus can be given in a number of forms.
1. A **polynomial**.
The base ring can be `\mathbb{Z}`, `\mathbb{Q}`, `\mathbb{Z}_p`, `\mathbb{F}_p`, or anything that can be converted to `\mathbb{Z}_p`.::
sage: P.<x> = ZZ[] sage: R.<a> = Zq(27, modulus = x^3 + 2*x + 1); R.modulus() (1 + O(3^20))*x^3 + (O(3^20))*x^2 + (2 + O(3^20))*x + (1 + O(3^20)) sage: P.<x> = QQ[] sage: S.<a> = Zq(27, modulus = x^3 + 2/7*x + 1) sage: P.<x> = Zp(3)[] sage: T.<a> = Zq(27, modulus = x^3 + 2*x + 1) sage: P.<x> = Qp(3)[] sage: U.<a> = Zq(27, modulus = x^3 + 2*x + 1) sage: P.<x> = GF(3)[] sage: V.<a> = Zq(27, modulus = x^3 + 2*x + 1)
Which form the modulus is given in has no effect on the unramified extension produced::
sage: R == S, R == T, T == U, U == V (False, True, True, False)
unless the modulus is different, or the precision of the modulus differs. In the case of ``V``, the modulus is only given to precision ``1``, so the resulting field has a precision cap of ``1``.::
sage: V.precision_cap() 1 sage: U.precision_cap() 20 sage: P.<x> = Zp(3)[] sage: modulus = x^3 + (2 + O(3^7))*x + (1 + O(3^10)) sage: modulus (1 + O(3^20))*x^3 + (2 + O(3^7))*x + (1 + O(3^10)) sage: W.<a> = Zq(27, modulus = modulus); W.precision_cap() 7
2. The modulus can also be given as a **symbolic expression**.::
sage: x = var('x') sage: X.<a> = Zq(27, modulus = x^3 + 2*x + 1); X.modulus() (1 + O(3^20))*x^3 + (O(3^20))*x^2 + (2 + O(3^20))*x + (1 + O(3^20)) sage: X == R True
By default, the polynomial chosen is the standard lift of the generator chosen for `\mathbb{F}_q`.::
sage: GF(125, 'a').modulus() x^3 + 3*x + 3 sage: Y.<a> = Zq(125); Y.modulus() (1 + O(5^20))*x^3 + (O(5^20))*x^2 + (3 + O(5^20))*x + (3 + O(5^20))
However, you can choose another polynomial if desired (as long as the reduction to `\mathbb{F}_p[x]` is irreducible).::
sage: P.<x> = ZZ[] sage: Z.<a> = Zq(125, modulus = x^3 + 3*x^2 + x + 1); Z.modulus() (1 + O(5^20))*x^3 + (3 + O(5^20))*x^2 + (1 + O(5^20))*x + (1 + O(5^20)) sage: Y == Z False
PRINTING:
There are many different ways to print `p`-adic elements. The way elements of a given field print is controlled by options passed in at the creation of the field. There are four basic printing modes (``'series'``, ``'val-unit'``, ``'terse'`` and ``'bars'``; ``'digits'`` is not available), as well as various options that either hide some information in the print representation or sometimes make print representations more compact. Note that the printing options affect whether different `p`-adic fields are considered equal.
1. **series**: elements are displayed as series in `p`.::
sage: R.<a> = Zq(9, 20, 'capped-rel', print_mode='series'); (1+2*a)^4 2 + (2*a + 2)*3 + (2*a + 1)*3^2 + O(3^20) sage: -3*(1+2*a)^4 3 + a*3^2 + 3^3 + (2*a + 2)*3^4 + (2*a + 2)*3^5 + (2*a + 2)*3^6 + (2*a + 2)*3^7 + (2*a + 2)*3^8 + (2*a + 2)*3^9 + (2*a + 2)*3^10 + (2*a + 2)*3^11 + (2*a + 2)*3^12 + (2*a + 2)*3^13 + (2*a + 2)*3^14 + (2*a + 2)*3^15 + (2*a + 2)*3^16 + (2*a + 2)*3^17 + (2*a + 2)*3^18 + (2*a + 2)*3^19 + (2*a + 2)*3^20 + O(3^21) sage: b = ~(3*a+18); b (a + 2)*3^-1 + 1 + 2*3 + (a + 1)*3^2 + 3^3 + 2*3^4 + (a + 1)*3^5 + 3^6 + 2*3^7 + (a + 1)*3^8 + 3^9 + 2*3^10 + (a + 1)*3^11 + 3^12 + 2*3^13 + (a + 1)*3^14 + 3^15 + 2*3^16 + (a + 1)*3^17 + 3^18 + O(3^19) sage: b.parent() is R.fraction_field() True
*print_pos* controls whether negatives can be used in the coefficients of powers of `p`.::
sage: S.<b> = Zq(9, print_mode='series', print_pos=False); (1+2*b)^4 -1 - b*3 - 3^2 + (b + 1)*3^3 + O(3^20) sage: -3*(1+2*b)^4 3 + b*3^2 + 3^3 + (-b - 1)*3^4 + O(3^21)
*ram_name* controls how the prime is printed.::
sage: T.<d> = Zq(9, print_mode='series', ram_name='p'); 3*(1+2*d)^4 2*p + (2*d + 2)*p^2 + (2*d + 1)*p^3 + O(p^21)
*print_max_ram_terms* limits the number of powers of `p` that appear.::
sage: U.<e> = Zq(9, print_mode='series', print_max_ram_terms=4); repr(-3*(1+2*e)^4) '3 + e*3^2 + 3^3 + (2*e + 2)*3^4 + ... + O(3^21)'
*print_max_unram_terms* limits the number of terms that appear in a coefficient of a power of `p`.::
sage: V.<f> = Zq(128, prec = 8, print_mode='series'); repr((1+f)^9) '(f^3 + 1) + (f^5 + f^4 + f^3 + f^2)*2 + (f^6 + f^5 + f^4 + f + 1)*2^2 + (f^5 + f^4 + f^2 + f + 1)*2^3 + (f^6 + f^5 + f^4 + f^3 + f^2 + f + 1)*2^4 + (f^5 + f^4)*2^5 + (f^6 + f^5 + f^4 + f^3 + f + 1)*2^6 + (f + 1)*2^7 + O(2^8)' sage: V.<f> = Zq(128, prec = 8, print_mode='series', print_max_unram_terms = 3); repr((1+f)^9) '(f^3 + 1) + (f^5 + f^4 + ... + f^2)*2 + (f^6 + f^5 + ... + 1)*2^2 + (f^5 + f^4 + ... + 1)*2^3 + (f^6 + f^5 + ... + 1)*2^4 + (f^5 + f^4)*2^5 + (f^6 + f^5 + ... + 1)*2^6 + (f + 1)*2^7 + O(2^8)' sage: V.<f> = Zq(128, prec = 8, print_mode='series', print_max_unram_terms = 2); repr((1+f)^9) '(f^3 + 1) + (f^5 + ... + f^2)*2 + (f^6 + ... + 1)*2^2 + (f^5 + ... + 1)*2^3 + (f^6 + ... + 1)*2^4 + (f^5 + f^4)*2^5 + (f^6 + ... + 1)*2^6 + (f + 1)*2^7 + O(2^8)' sage: V.<f> = Zq(128, prec = 8, print_mode='series', print_max_unram_terms = 1); repr((1+f)^9) '(f^3 + ...) + (f^5 + ...)*2 + (f^6 + ...)*2^2 + (f^5 + ...)*2^3 + (f^6 + ...)*2^4 + (f^5 + ...)*2^5 + (f^6 + ...)*2^6 + (f + ...)*2^7 + O(2^8)' sage: V.<f> = Zq(128, prec = 8, print_mode='series', print_max_unram_terms = 0); repr((1+f)^9 - 1 - f^3) '(...)*2 + (...)*2^2 + (...)*2^3 + (...)*2^4 + (...)*2^5 + (...)*2^6 + (...)*2^7 + O(2^8)'
*show_prec* determines whether the precision is printed (default ``True``)::
sage: U.<e> = Zq(9, 2, show_prec=False); repr(-3*(1+2*e)^4) '3 + e*3^2'
*print_sep* and *print_max_terse_terms* have no effect.
Note that print options affect equality::
sage: R == S, R == T, R == U, R == V, S == T, S == U, S == V, T == U, T == V, U == V (False, False, False, False, False, False, False, False, False, False)
2. **val-unit**: elements are displayed as `p^k u`::
sage: R.<a> = Zq(9, 7, print_mode='val-unit'); b = (1+3*a)^9 - 1; b 3^3 * (15 + 64*a) + O(3^7) sage: ~b 3^-3 * (41 + a) + O(3)
*print_pos* controls whether to use a balanced representation or not.::
sage: S.<a> = Zq(9, 7, print_mode='val-unit', print_pos=False); b = (1+3*a)^9 - 1; b 3^3 * (15 - 17*a) + O(3^7) sage: ~b 3^-3 * (-40 + a) + O(3)
*ram_name* affects how the prime is printed.::
sage: A.<x> = Zp(next_prime(10^6), print_mode='val-unit')[] sage: T.<a> = Zq(next_prime(10^6)^3, 4, print_mode='val-unit', ram_name='p', modulus=x^3+385831*x^2+106556*x+321036); b = (next_prime(10^6)^2*(a^2 + a - 4)^4); b p^2 * (87996187118837557387483 + 246348888344392418464080*a + 1353538653775332610349*a^2) + O(p^6) sage: b * (a^2 + a - 4)^-4 p^2 * 1 + O(p^6)
*print_max_terse_terms* controls how many terms of the polynomial appear in the unit part.::
sage: U.<a> = Zq(17^4, 6, print_mode='val-unit', print_max_terse_terms=3); b = (17*(a^3-a+14)^6); b 17 * (12131797 + 12076378*a + 10809706*a^2 + ...) + O(17^7)
*show_prec* determines whether the precision is printed (default ``True``)::
sage: U.<e> = Zq(9, 2, print_mode='val-unit', show_prec=False); repr(-3*(1+2*e)^4) '3 * (1 + 3*e)'
*print_sep*, *print_max_ram_terms* and *print_max_unram_terms* have no effect.
Equality again depends on the printing options::
sage: R == S, R == T, R == U, S == T, S == U, T == U (False, False, False, False, False, False)
3. **terse**: elements are displayed as a polynomial of degree less than the degree of the extension.::
sage: R.<a> = Zq(125, print_mode='terse') sage: (a+5)^177 68210977979428 + 90313850704069*a + 73948093055069*a^2 + O(5^20) sage: (a/5+1)^177 68210977979428/5^177 + 90313850704069/5^177*a + 73948093055069/5^177*a^2 + O(5^-157)
Note that in this last computation, you get one fewer `p`-adic digit than one might expect. This is because ``R`` is capped absolute, and thus 5 is cast in with relative precision 19.
As of version 3.3, if coefficients of the polynomial are non-integral, they are always printed with an explicit power of `p` in the denominator.::
sage: 5*a + a^2/25 5*a + 1/5^2*a^2 + O(5^18)
*print_pos* controls whether to use a balanced representation or not.::
sage: (a-5)^6 22864 + 95367431627998*a + 8349*a^2 + O(5^20) sage: S.<a> = Zq(125, print_mode='terse', print_pos=False); b = (a-5)^6; b 22864 - 12627*a + 8349*a^2 + O(5^20) sage: (a - 1/5)^6 -20624/5^6 + 18369/5^5*a + 1353/5^3*a^2 + O(5^14)
*ram_name* affects how the prime is printed.::
sage: T.<a> = Zq(125, print_mode='terse', ram_name='p'); (a - 1/5)^6 95367431620001/p^6 + 18369/p^5*a + 1353/p^3*a^2 + O(p^14)
*print_max_terse_terms* controls how many terms of the polynomial are shown.::
sage: U.<a> = Zq(625, print_mode='terse', print_max_terse_terms=2); (a-1/5)^6 106251/5^6 + 49994/5^5*a + ... + O(5^14)
*show_prec* determines whether the precision is printed (default ``True``)::
sage: U.<e> = Zq(9, 2, print_mode='terse', show_prec=False); repr(-3*(1+2*e)^4) '3 + 9*e'
*print_sep*, *print_max_ram_terms* and *print_max_unram_terms* have no effect.
Equality again depends on the printing options::
sage: R == S, R == T, R == U, S == T, S == U, T == U (False, False, False, False, False, False)
4. **digits**: This print mode is not available when the residue field is not prime. It might make sense to have a dictionary for small fields, but this isn't implemented.
5. **bars**: elements are displayed in a similar fashion to series, but more compactly.::
sage: R.<a> = Zq(125); (a+5)^6 (4*a^2 + 3*a + 4) + (3*a^2 + 2*a)*5 + (a^2 + a + 1)*5^2 + (3*a + 2)*5^3 + (3*a^2 + a + 3)*5^4 + (2*a^2 + 3*a + 2)*5^5 + O(5^20) sage: R.<a> = Zq(125, print_mode='bars', prec=8); repr((a+5)^6) '...[2, 3, 2]|[3, 1, 3]|[2, 3]|[1, 1, 1]|[0, 2, 3]|[4, 3, 4]' sage: repr((a-5)^6) '...[0, 4]|[1, 4]|[2, 0, 2]|[1, 4, 3]|[2, 3, 1]|[4, 4, 3]|[2, 4, 4]|[4, 3, 4]'
Note that it's not possible to read of the precision from the representation in this mode.::
sage: b = a + 3; repr(b) '...[3, 1]' sage: c = a + R(3, 4); repr(c) '...[3, 1]' sage: b.precision_absolute() 8 sage: c.precision_absolute() 4
*print_pos* controls whether the digits can be negative.::
sage: S.<a> = Zq(125, print_mode='bars', print_pos=False); repr((a-5)^6) '...[1, -1, 1]|[2, 1, -2]|[2, 0, -2]|[-2, -1, 2]|[0, 0, -1]|[-2]|[-1, -2, -1]' sage: repr((a-1/5)^6) '...[0, 1, 2]|[-1, 1, 1]|.|[-2, -1, -1]|[2, 2, 1]|[0, 0, -2]|[0, -1]|[0, -1]|[1]'
*print_max_ram_terms* controls the maximum number of "digits" shown. Note that this puts a cap on the relative precision, not the absolute precision.::
sage: T.<a> = Zq(125, print_mode='bars', print_max_ram_terms=3, print_pos=False); repr((a-5)^6) '...[0, 0, -1]|[-2]|[-1, -2, -1]' sage: repr(5*(a-5)^6+50) '...[0, 0, -1]|[]|[-1, -2, -1]|[]'
However, if the element has negative valuation, digits are shown up to the decimal point.::
sage: repr((a-1/5)^6) '...|.|[-2, -1, -1]|[2, 2, 1]|[0, 0, -2]|[0, -1]|[0, -1]|[1]'
*print_sep* controls the separating character (``'|'`` by default).::
sage: U.<a> = Zq(625, print_mode='bars', print_sep=''); b = (a+5)^6; repr(b) '...[0, 1][4, 0, 2][3, 2, 2, 3][4, 2, 2, 4][0, 3][1, 1, 3][3, 1, 4, 1]'
*print_max_unram_terms* controls how many terms are shown in each ``'digit'``::
sage: with local_print_mode(U, {'max_unram_terms': 3}): repr(b) '...[0, 1][4,..., 0, 2][3,..., 2, 3][4,..., 2, 4][0, 3][1,..., 1, 3][3,..., 4, 1]' sage: with local_print_mode(U, {'max_unram_terms': 2}): repr(b) '...[0, 1][4,..., 2][3,..., 3][4,..., 4][0, 3][1,..., 3][3,..., 1]' sage: with local_print_mode(U, {'max_unram_terms': 1}): repr(b) '...[..., 1][..., 2][..., 3][..., 4][..., 3][..., 3][..., 1]' sage: with local_print_mode(U, {'max_unram_terms':0}): repr(b-75*a) '...[...][...][...][...][][...][...]'
*show_prec* determines whether the precision is printed (default ``False``)::
sage: U.<e> = Zq(9, 2, print_mode='bars', show_prec=True); repr(-3*(1+2*e)^4) '[0, 1]|[1]|[] + O(3^3)'
*ram_name* and *print_max_terse_terms* have no effect.
Equality depends on printing options::
sage: R == S, R == T, R == U, S == T, S == U, T == U (False, False, False, False, False, False)
EXAMPLES
Unlike for ``Zp``, you can't create ``Zq(N)`` when ``N`` is not a prime power.
However, you can use ``check=False`` to pass in a pair in order to not have to factor. If you do so, you need to use names explicitly rather than the ``R.<a>`` syntax.::
sage: p = next_prime(2^123) sage: k = Zp(p) sage: R.<x> = k[] sage: K = Zq([(p, 5)], modulus=x^5+x+4, names='a', ram_name='p', print_pos=False, check=False) sage: K.0^5 (-a - 4) + O(p^20)
In tests on sage.math, the creation of ``K`` as above took an average of 1.58ms, while::
sage: K = Zq(p^5, modulus=x^5+x+4, names='a', ram_name='p', print_pos=False, check=True)
took an average of 24.5ms. Of course, with smaller primes these savings disappear.
TESTS::
sage: R = Zq([(5,3)], names="alpha"); R Unramified Extension in alpha defined by x^3 + 3*x + 3 with capped relative precision 20 over 5-adic Ring sage: Zq((5, 3), names="alpha") is R True sage: Zq(125.factor(), names="alpha") is R True
""" else: raise ValueError("q must be a prime power") raise ValueError("q must have shape [(p, k)]") raise ValueError("q must be a prime power") else: raise ValueError("q must be a prime power") prec = Integer(prec) raise TypeError("modulus must be a polynomial") names = str(names) #raise TypeError, "names must be a string" raise ValueError("q must be a prime power") else: print_pos=print_pos, print_sep=print_sep, print_max_terms=print_max_ram_terms, show_prec=show_prec, check=False) return base raise TypeError("You must specify the name of the generator.") names=names, res_name=res_name, ram_name=ram_name, print_pos=print_pos, print_sep=print_sep, print_max_ram_terms=print_max_ram_terms, print_max_unram_terms=print_max_unram_terms, print_max_terse_terms=print_max_terse_terms, show_prec=show_prec, check=check, unram=True, implementation=implementation)
###################################################### # Short constructor names for different types ######################################################
def ZpCR(p, prec = None, *args, **kwds): r""" A shortcut function to create capped relative `p`-adic rings.
Same functionality as ``Zp``. See documentation for ``Zp`` for a description of the input parameters.
EXAMPLES::
sage: ZpCR(5, 40) 5-adic Ring with capped relative precision 40 """
def ZpCA(p, prec = None, *args, **kwds): r""" A shortcut function to create capped absolute `p`-adic rings.
See documentation for :func:`Zp` for a description of the input parameters.
EXAMPLES::
sage: ZpCA(5, 40) 5-adic Ring with capped absolute precision 40 """
def ZpFM(p, prec = None, *args, **kwds): r""" A shortcut function to create fixed modulus `p`-adic rings.
See documentation for :func:`Zp` for a description of the input parameters.
EXAMPLES::
sage: ZpFM(5, 40) 5-adic Ring of fixed modulus 5^40 """
def ZpFP(p, prec = None, *args, **kwds): r""" A shortcut function to create floating point `p`-adic rings.
Same functionality as ``Zp``. See documentation for ``Zp`` for a description of the input parameters.
EXAMPLES::
sage: ZpFP(5, 40) 5-adic Ring with floating precision 40 """
def ZqCR(q, prec = None, *args, **kwds): r""" A shortcut function to create capped relative unramified `p`-adic rings.
Same functionality as ``Zq``. See documentation for ``Zq`` for a description of the input parameters.
EXAMPLES::
sage: R.<a> = ZqCR(25, 40); R Unramified Extension in a defined by x^2 + 4*x + 2 with capped relative precision 40 over 5-adic Ring """
def ZqCA(q, prec = None, *args, **kwds): r""" A shortcut function to create capped absolute unramified `p`-adic rings.
See documentation for :func:`Zq` for a description of the input parameters.
EXAMPLES::
sage: R.<a> = ZqCA(25, 40); R Unramified Extension in a defined by x^2 + 4*x + 2 with capped absolute precision 40 over 5-adic Ring """
def ZqFM(q, prec = None, *args, **kwds): r""" A shortcut function to create fixed modulus unramified `p`-adic rings.
See documentation for :func:`Zq` for a description of the input parameters.
EXAMPLES::
sage: R.<a> = ZqFM(25, 40); R Unramified Extension in a defined by x^2 + 4*x + 2 of fixed modulus 5^40 over 5-adic Ring """
def ZqFP(q, prec = None, *args, **kwds): r""" A shortcut function to create floating point unramified `p`-adic rings.
Same functionality as ``Zq``. See documentation for ``Zq`` for a description of the input parameters.
EXAMPLES::
sage: R.<a> = ZqFP(25, 40); R Unramified Extension in a defined by x^2 + 4*x + 2 with floating precision 40 over 5-adic Ring """
@experimental(23505) def ZpLC(p, prec=None, *args, **kwds): r""" A shortcut function to create `p`-adic rings with lattice precision (precision is encoded by a lattice in a large vector space and tracked using automatic differentiation).
See documentation for :func:`Zp` for a description of the input parameters.
EXAMPLES:
Below is a small demo of the features by this model of precision::
sage: R = ZpLC(3, print_mode='terse') sage: R 3-adic Ring with lattice-cap precision
sage: x = R(1,10)
Of course, when we multiply by 3, we gain one digit of absolute precision::
sage: 3*x 3 + O(3^11)
The lattice precision machinery sees this even if we decompose the computation into several steps::
sage: y = x+x sage: y 2 + O(3^10) sage: x + y 3 + O(3^11)
The same works for the multiplication::
sage: z = x^2 sage: z 1 + O(3^10) sage: x*z 1 + O(3^11)
This can be more surprising when we are working with elements given at different precisions::
sage: R = ZpLC(2, print_mode='terse') sage: x = R(1,10) sage: y = R(1,5) sage: z = x+y; z 2 + O(2^5) sage: t = x-y; t 0 + O(2^5) sage: z+t # observe that z+t = 2*x 2 + O(2^11) sage: z-t # observe that z-t = 2*y 2 + O(2^6)
sage: x = R(28888,15) sage: y = R(204,10) sage: z = x/y; z 242 + O(2^9) sage: z*y # which is x 28888 + O(2^15)
The SOMOS sequence is the sequence defined by the recurrence:
..MATH::
u_n = \frac {u_{n-1} u_{n-3} + u_{n-2}^2} {u_{n-4}}
It is known for its numerical instability. On the one hand, one can show that if the initial values are invertible in `\mathbb{Z}_p` and known at precision `O(p^N)` then all the next terms of the SOMOS sequence will be known at the same precision as well. On the other hand, because of the division, when we unroll the recurrence, we loose a lot of precision. Observe::
sage: R = Zp(2, 30, print_mode='terse') sage: a,b,c,d = R(1,15), R(1,15), R(1,15), R(3,15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 4 + O(2^15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 13 + O(2^15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 55 + O(2^15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 21975 + O(2^15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 6639 + O(2^13) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 7186 + O(2^13) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 569 + O(2^13) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 253 + O(2^13) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 4149 + O(2^13) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 2899 + O(2^12) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 3072 + O(2^12) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 349 + O(2^12) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 619 + O(2^12) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 243 + O(2^12) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 3 + O(2^2) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 2 + O(2^2)
If instead, we use the lattice precision, everything goes well::
sage: R = ZpLC(2, 30, print_mode='terse') sage: a,b,c,d = R(1,15), R(1,15), R(1,15), R(3,15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 4 + O(2^15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 13 + O(2^15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 55 + O(2^15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 21975 + O(2^15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 23023 + O(2^15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 31762 + O(2^15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 16953 + O(2^15) sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print(d) 16637 + O(2^15)
sage: for _ in range(100): ....: a,b,c,d = b,c,d,(b*d+c*c)/a sage: a 15519 + O(2^15) sage: b 32042 + O(2^15) sage: c 17769 + O(2^15) sage: d 20949 + O(2^15)
ALGORITHM:
The precision is global. It is encoded by a lattice in a huge vector space whose dimension is the number of elements having this parent. Precision is tracked using automatic differentiation techniques (see [CRV2014]_ and [CRV2018]_).
Concretely, this precision datum is an instance of the class :class:`sage.rings.padic.lattice_precision.PrecisionLattice`. It is attached to the parent and is created at the same time as the parent. (It is actually a bit more subtle because two different parents may share the same instance; this happens for instance for a `p`-adic ring and its field of fractions.)
This precision datum is accessible through the method :meth:`precision`::
sage: R = ZpLC(5, print_mode='terse') sage: prec = R.precision() sage: prec Precision lattice on 0 objects
This instance knows about all elements of the parent. It is automatically updated when a new element (of this parent) is created::
sage: x = R(3513,10) sage: prec Precision lattice on 1 object sage: y = R(176,5) sage: prec Precision lattice on 2 objects sage: z = R.random_element() sage: prec Precision lattice on 3 objects
The method :meth:`tracked_elements` provides the list of all tracked elements::
sage: prec.tracked_elements() [3513 + O(5^10), 176 + O(5^5), ...]
Similarly, when a variable is collected by the garbage collector, the precision lattice is updated. Note however that the update might be delayed. We can force it with the method :meth:`del_elements`::
sage: z = 0 sage: prec # random output, could be 2 objects if the garbage collector is fast Precision lattice on 3 objects sage: prec.del_elements() sage: prec Precision lattice on 2 objects
The method :meth:`precision_lattice` returns (a matrix defining) the lattice that models the precision. Here we have::
sage: prec.precision_lattice() [9765625 0] [ 0 3125]
Observe that `5^10 = 9765625` and `5^5 = 3125`. The above matrix then reflects the precision on `x` and `y`.
Now, observe how the precision lattice changes while performing computations::
sage: x, y = 3*x+2*y, 2*(x-y) sage: prec.del_elements() sage: prec.precision_lattice() [ 3125 48825000] [ 0 48828125]
The matrix we get is no longer diagonal, meaning that some digits of precision are diffused among the two new elements `x` and `y`. They nevertheless show up when we compute for instance `x+y`::
sage: x 1516 + O(5^5) sage: y 424 + O(5^5) sage: x+y 17565 + O(5^11)
These diffused digits of precision (which are tracked but do not appear on the printing) allow to be always sharp on precision.
NOTE:
Each elementary operation requires significant manipulations on the precision lattice and therefore is costly. Precisely:
- The creation of a new element has a cost `O(n)` where `n` is the number of tracked elements.
- The destruction of one element has a cost `O(m^2)` where `m` is the distance between the destroyed element and the last one. Fortunately, it seems that `m` tends to be small in general (the dynamics of the list of tracked elements is rather close to that of a stack).
It is nevertheless still possible to manipulate several hundred variables (e.g. square matrices of size 5 or polynomials of degree 20).
The class :class:`PrecisionLattice` provides several features for introspection, especially concerning timings. See :meth:`history` and :meth:`timings` for details.
.. SEEALSO::
:func:`ZpLF` """
@experimental(23505) def ZpLF(p, prec=None, *args, **kwds): r""" A shortcut function to create `p`-adic rings where precision is encoded by a module in a large vector space.
See documentation for :func:`Zp` for a description of the input parameters.
NOTE:
The precision is tracked using automatic differentiation techniques (see [CRV2018]_ and [CRV2014]_). Floating point `p`-adic numbers are used for the computation of the differential (which is then not exact).
EXAMPLES::
sage: R = ZpLF(5, 40) sage: R 5-adic Ring with lattice-float precision
.. SEEALSO::
:func:`ZpLC` """
####################################################################################################### # # The Extension Factory -- creates extensions of p-adic rings and fields # #######################################################################################################
class pAdicExtension_class(UniqueFactory): r""" A class for creating extensions of `p`-adic rings and fields.
EXAMPLES::
sage: R = Zp(5,3) sage: S.<x> = ZZ[] sage: W.<w> = pAdicExtension(R, x^4-15) sage: W Eisenstein Extension in w defined by x^4 - 15 with capped relative precision 12 over 5-adic Ring sage: W.precision_cap() 12 """ def create_key_and_extra_args(self, base, modulus, prec = None, print_mode = None, names = None, var_name = None, res_name = None, unram_name = None, ram_name = None, print_pos = None, print_sep = None, print_alphabet = None, print_max_ram_terms = None, print_max_unram_terms = None, print_max_terse_terms = None, show_prec = None, check = True, unram = False, implementation='FLINT'): r""" Creates a key from input parameters for pAdicExtension.
See the documentation for ``Qq`` for more information.
TESTS::
sage: R = Zp(5,3) sage: S.<x> = ZZ[] sage: pAdicExtension.create_key_and_extra_args(R, x^4-15,names='w') (('e', 5-adic Ring with capped relative precision 3, x^4 - 15, ('w', None, None, 'w'), 12, 'series', True, '|', (), -1, -1, -1, True, 'NTL'), {'approx_modulus': (1 + O(5^3))*x^4 + (O(5^4))*x^3 + (O(5^4))*x^2 + (O(5^4))*x + (2*5 + 4*5^2 + 4*5^3 + O(5^4))})
sage: A = Qp(3,5) sage: Po.<X> = A[] sage: f = Po([3,0,-1]) sage: K.<a> = A.ext(f) sage: -a^2+3 O(a^12) sage: K.defining_polynomial() == f/f.leading_coefficient() True
sage: g = Po([6,3,2]) sage: H.<b> = A.ext(g) sage: 2*b^2+3*b+6 O(b^12) sage: H.defining_polynomial() == g/g.leading_coefficient() True """ raise ValueError("symbolic expression must be in only one variable") raise ValueError("must use univariate polynomial") else: raise ValueError("modulus must be a polynomial") raise NotImplementedError("degree of modulus must be at least 2") # need to add more checking here. if var_name is not None: names = var_name else: raise ValueError("must specify name of generator of extension") names = str(names) else:
# We now decide on the extension class: unramified, Eisenstein, two-step or general raise ValueError("Precision cannot be larger than that of base ring; you may want to call the change method on the base ring.")
raise ValueError("Precision cannot be larger than that of base ring; you may want to call the change method on the base ring.") else: print_sep, tuple(print_alphabet), print_max_ram_terms, print_max_unram_terms, print_max_terse_terms, show_prec, implementation)
def create_object(self, version, key, approx_modulus=None, shift_seed=None): r""" Creates an object using a given key.
See the documentation for pAdicExtension for more information.
TESTS::
sage: R = Zp(5,3) sage: S.<x> = R[] sage: pAdicExtension.create_object(version = (6,4,2), key = ('e', R, x^4 - 15, x^4 - 15, ('w', None, None, 'w'), 12, None, 'series', True, '|', (),-1,-1,-1,'NTL'), shift_seed = S(3 + O(5^3))) Eisenstein Extension in w defined by x^4 - 15 with capped relative precision 12 over 5-adic Ring """ key = list(key) key.append('NTL') print_alphabet, print_max_ram_terms, print_max_unram_terms, print_max_terse_terms, implementation) = key exact_modulus = premodulus.polynomial(base.exact_field()) else: print_sep, print_alphabet, print_max_ram_terms, print_max_unram_terms, print_max_terse_terms, show_prec, implementation) = key raise ValueError("polynomial does not determine a unique extension. Please specify more precision or use parameter check=False.")
{'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet, 'max_ram_terms': print_max_ram_terms, 'max_unram_terms': print_max_unram_terms, 'max_terse_terms': print_max_terse_terms, 'show_prec': show_prec}, shift_seed, names, implementation)
ExtensionFactory = pAdicExtension = pAdicExtension_class("pAdicExtension")
###################################################### # Helper functions for the Extension Factory ######################################################
def split(poly, prec): r""" Given a polynomial ``poly`` and a desired precision ``prec``, computes ``upoly`` and epoly so that the extension defined by ``poly`` is isomorphic to the extension defined by first taking an extension by the unramified polynomial ``upoly``, and then an extension by the Eisenstein polynomial ``epoly``.
We need better `p`-adic factoring in Sage before this function can be implemented.
EXAMPLES::
sage: k = Qp(13) sage: x = polygen(k) sage: f = x^2+1 sage: sage.rings.padics.factory.split(f, 10) Traceback (most recent call last): ... NotImplementedError: Extensions by general polynomials not yet supported. Please use an unramified or Eisenstein polynomial.
TESTS:
This checks that :trac:`6186` is still fixed::
sage: k = Qp(13) sage: x = polygen(k) sage: f = x^2+1 sage: L.<a> = k.extension(f) Traceback (most recent call last): ... NotImplementedError: Extensions by general polynomials not yet supported. Please use an unramified or Eisenstein polynomial.
"""
def truncate_to_prec(poly, R, absprec): r""" Truncates the unused precision off of a polynomial.
EXAMPLES::
sage: R = Zp(5) sage: S.<x> = R[] sage: from sage.rings.padics.factory import truncate_to_prec sage: f = x^4 + (3+O(5^6))*x^3 + O(5^4) sage: truncate_to_prec(f, R, 5) (1 + O(5^5))*x^4 + (3 + O(5^5))*x^3 + (O(5^5))*x^2 + (O(5^5))*x + (O(5^4)) """
def krasner_check(poly, prec): r""" Returns True iff poly determines a unique isomorphism class of extensions at precision prec.
Currently just returns True (thus allowing extensions that are not defined to high enough precision in order to specify them up to isomorphism). This will change in the future.
EXAMPLES::
sage: from sage.rings.padics.factory import krasner_check sage: krasner_check(1,2) #this is a stupid example. True """
def is_eisenstein(poly): r""" Returns True iff this monic polynomial is Eisenstein.
A polynomial is Eisenstein if it is monic, the constant term has valuation 1 and all other terms have positive valuation.
EXAMPLES::
sage: R = Zp(5) sage: S.<x> = R[] sage: from sage.rings.padics.factory import is_eisenstein sage: f = x^4 - 75*x + 15 sage: is_eisenstein(f) True sage: g = x^4 + 75 sage: is_eisenstein(g) False sage: h = x^7 + 27*x -15 sage: is_eisenstein(h) False """
def is_unramified(poly): r""" Returns true iff this monic polynomial is unramified.
A polynomial is unramified if its reduction modulo the maximal ideal is irreducible.
EXAMPLES::
sage: R = Zp(5) sage: S.<x> = R[] sage: from sage.rings.padics.factory import is_unramified sage: f = x^4 + 14*x + 9 sage: is_unramified(f) True sage: g = x^6 + 17*x + 6 sage: is_unramified(g) False """ return False |