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# -*- coding: utf-8 -*- """ Function Fields
AUTHORS:
- William Stein (2010): initial version
- Robert Bradshaw (2010-05-30): added is_finite()
- Julian Rüth (2011-06-08, 2011-09-14, 2014-06-23, 2014-06-24, 2016-11-13): fixed hom(), extension(); use @cached_method; added derivation(); added support for relative vector spaces; fixed conversion to base fields
- Maarten Derickx (2011-09-11): added doctests
- Syed Ahmad Lavasani (2011-12-16): added genus(), is_RationalFunctionField()
- Simon King (2014-10-29): Use the same generator names for a function field extension and the underlying polynomial ring.
EXAMPLES:
We create an extension of a rational function fields, and do some simple arithmetic in it::
sage: K.<x> = FunctionField(GF(5^2,'a')); K Rational function field in x over Finite Field in a of size 5^2 sage: R.<y> = K[] sage: L.<y> = K.extension(y^3 - (x^3 + 2*x*y + 1/x)); L Function field in y defined by y^3 + 3*x*y + (4*x^4 + 4)/x sage: y^2 y^2 sage: y^3 2*x*y + (x^4 + 1)/x sage: a = 1/y; a (4*x/(4*x^4 + 4))*y^2 + 2*x^2/(4*x^4 + 4) sage: a * y 1
We next make an extension of the above function field, illustrating that arithmetic with a tower of 3 fields is fully supported::
sage: S.<t> = L[] sage: M.<t> = L.extension(t^2 - x*y) sage: M Function field in t defined by t^2 + 4*x*y sage: t^2 x*y sage: 1/t ((1/(x^4 + 1))*y^2 + 2*x/(4*x^4 + 4))*t sage: M.base_field() Function field in y defined by y^3 + 3*x*y + (4*x^4 + 4)/x sage: M.base_field().base_field() Rational function field in x over Finite Field in a of size 5^2
It is also possible to construct function fields over an imperfect base field::
sage: N.<u> = FunctionField(K)
and function fields as inseparable extensions::
sage: R.<v> = K[] sage: O.<v> = K.extension(v^5 - x)
TESTS::
sage: TestSuite(K).run(max_runs=1024) # long time (5s) sage: TestSuite(L).run(max_runs=64) # long time (10s) sage: TestSuite(M).run(max_runs=32) # long time (30s) sage: TestSuite(N).run(max_runs=64, skip = '_test_derivation') # long time (8s) sage: TestSuite(O).run(max_runs=128, skip = '_test_derivation') # long time (8s)
sage: TestSuite(R).run() sage: TestSuite(S).run() # long time (3s) """ from __future__ import absolute_import #***************************************************************************** # Copyright (C) 2010 William Stein <wstein@gmail.com> # Copyright (C) 2010 Robert Bradshaw <robertwb@math.washington.edu> # Copyright (C) 2011-2017 Julian Rüth <julian.rueth@gmail.com> # Copyright (C) 2011 Maarten Derickx <m.derickx.student@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 sage.rings.ring import Field from .function_field_element import FunctionFieldElement, FunctionFieldElement_rational, FunctionFieldElement_polymod
from sage.misc.cachefunc import cached_method
#is needed for genus computation from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing from sage.interfaces.all import singular
from sage.categories.function_fields import FunctionFields CAT = FunctionFields()
def is_FunctionField(x): """ Return True if ``x`` is of function field type.
EXAMPLES::
sage: from sage.rings.function_field.function_field import is_FunctionField sage: is_FunctionField(QQ) False sage: is_FunctionField(FunctionField(QQ,'t')) True """
class FunctionField(Field): """ The abstract base class for all function fields.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: isinstance(K, sage.rings.function_field.function_field.FunctionField) True """ def __init__(self, base_field, names, category = CAT): r""" TESTS::
sage: K.<x> = FunctionField(QQ) sage: from sage.rings.function_field.function_field import FunctionField sage: isinstance(K, FunctionField) True
"""
# allow conversion into the constant base field # the conversion map must not keep this field alive if that is the only reference to it
def is_perfect(self): r""" Return whether this field is perfect, i.e., its characteristic is `p=0` or every element has a `p`-th root.
EXAMPLES::
sage: FunctionField(QQ, 'x').is_perfect() True sage: FunctionField(GF(2), 'x').is_perfect() False
"""
def some_elements(self): """ Return some elemnts in this function field.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: K.some_elements() [1, x, 2*x, x/(x^2 + 2*x + 1), 1/x^2, x/(x^2 - 1), x/(x^2 + 1), x/(2*x^2 + 2), 0, 1/x, ...]
::
sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: L.some_elements() [1, y, 1/x*y, ((1/4*x + 1/4)/(1/4*x^2 - 1/2*x + 1/4))*y - 1/2*x/(1/4*x^2 - 1/2*x + 1/4), -1/-x, (1/(x - 1))*y, (1/(x + 1))*y, (1/(2*x + 2))*y, 0, ...]
"""
def characteristic(self): """ Return the characteristic of this function field.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: K.characteristic() 0 sage: K.<x> = FunctionField(GF(7)) sage: K.characteristic() 7 sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: L.characteristic() 7 """
def is_finite(self): """ Return whether this function field is finite, which it is not.
EXAMPLES::
sage: R.<t> = FunctionField(QQ) sage: R.is_finite() False sage: R.<t> = FunctionField(GF(7)) sage: R.is_finite() False """
def extension(self, f, names=None): """ Create an extension L = K[y]/(f(y)) of a function field, defined by a univariate polynomial in one variable over this function field K.
INPUT:
- ``f`` -- a univariate polynomial over self - ``names`` -- None or string or length-1 tuple
OUTPUT:
- a function field
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: K.extension(y^5 - x^3 - 3*x + x*y) Function field in y defined by y^5 + x*y - x^3 - 3*x
A nonintegral defining polynomial::
sage: K.<t> = FunctionField(QQ); R.<y> = K[] sage: K.extension(y^3 + (1/t)*y + t^3/(t+1)) Function field in y defined by y^3 + 1/t*y + t^3/(t + 1)
The defining polynomial need not be monic or integral::
sage: K.extension(t*y^3 + (1/t)*y + t^3/(t+1)) Function field in y defined by t*y^3 + 1/t*y + t^3/(t + 1) """
def order_with_basis(self, basis, check=True): """ Return the order with given basis over the maximal order of the base field.
INPUT:
- ``basis`` -- a list of elements of self - ``check`` -- bool (default: True); if True, check that the basis is really linearly independent and that the module it spans is closed under multiplication, and contains the identity element.
OUTPUT:
- an order in this function field
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[]; L.<y> = K.extension(y^3 + x^3 + 4*x + 1) sage: O = L.order_with_basis([1, y, y^2]); O Order in Function field in y defined by y^3 + x^3 + 4*x + 1 sage: O.basis() (1, y, y^2)
Note that 1 does not need to be an element of the basis, as long it is in the module spanned by it::
sage: O = L.order_with_basis([1+y, y, y^2]); O Order in Function field in y defined by y^3 + x^3 + 4*x + 1 sage: O.basis() (y + 1, y, y^2)
The following error is raised when the module spanned by the basis is not closed under multiplication::
sage: O = L.order_with_basis([1, x^2 + x*y, (2/3)*y^2]); O Traceback (most recent call last): ... ValueError: The module generated by basis [1, x*y + x^2, 2/3*y^2] must be closed under multiplication
and this happens when the identity is not in the module spanned by the basis::
sage: O = L.order_with_basis([x, x^2 + x*y, (2/3)*y^2]) Traceback (most recent call last): ... ValueError: The identity element must be in the module spanned by basis [x, x*y + x^2, 2/3*y^2] """
def order(self, x, check=True): """ Return the order in this function field generated over the maximal order by x or the elements of x if x is a list.
INPUT:
- ``x`` -- element of self, or a list of elements of self - ``check`` -- bool (default: True); if True, check that x really generates an order
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[]; L.<y> = K.extension(y^3 + x^3 + 4*x + 1) sage: O = L.order(y); O Order in Function field in y defined by y^3 + x^3 + 4*x + 1 sage: O.basis() (1, y, y^2)
sage: Z = K.order(x); Z Order in Rational function field in x over Rational Field sage: Z.basis() (1,)
Orders with multiple generators, not yet supported::
sage: Z = K.order([x,x^2]); Z Traceback (most recent call last): ... NotImplementedError """ else:
def _coerce_map_from_(self, source): """ Return True if there is a coerce map from R to self.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[]; L.<y> = K.extension(y^3 + x^3 + 4*x + 1) sage: L.equation_order() Order in Function field in y defined by y^3 + x^3 + 4*x + 1 sage: L._coerce_map_from_(L.equation_order()) Conversion map: From: Order in Function field in y defined by y^3 + x^3 + 4*x + 1 To: Function field in y defined by y^3 + x^3 + 4*x + 1 sage: L._coerce_map_from_(GF(7))
sage: K.<x> = FunctionField(QQ) sage: L.<x> = FunctionField(GaussianIntegers().fraction_field()) sage: L.has_coerce_map_from(K) True
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^3 + 1) sage: K.<x> = FunctionField(GaussianIntegers().fraction_field()) sage: R.<y> = K[] sage: M.<y> = K.extension(y^3 + 1) sage: M.has_coerce_map_from(L) # not tested (the constant field including into a function field is not yet known to be injective) True
sage: K.<x> = FunctionField(QQ) sage: R.<I> = K[] sage: L.<I> = K.extension(I^2 + 1) sage: M.<x> = FunctionField(GaussianIntegers().fraction_field()) sage: M.has_coerce_map_from(L) True """ source_to_K = K.coerce_map_from(source) K_to_self = self.coerce_map_from(K) if source_to_K and K_to_self: return K_to_self * source_to_K # source and self are rational function fields # ... in the same variable else: # source is an extensions of rational function fields # the base field of source coerces into the base field of self # try to find a root of the defining polynomial in self # The defining polynomial of source has a root in self, # therefore there is a map. To be sure that it is # canonical, we require a root of the defining polynomial # of self to be a root of the defining polynomial of # source (and that the variables are named equally):
else: # The defining polynomial of source has a root in self, # therefore there is a map. To be sure that it is # canonical, we require the names of the roots to match
def _test_derivation(self, **options): """ Test the correctness of the derivations of the function field.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: TestSuite(K).run() # indirect doctest """ # Leibniz's law # Linearity # Constants map to zero tester.assertTrue(d(c) == 0)
def _convert_map_from_(self, R): r""" Return a conversion from ``R`` to this function field if one exists.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^3 + x^3 + 4*x + 1) sage: K(L(x)) # indirect doctest x
"""
def _intermediate_fields(self, base): r""" Return the fields which lie in between ``base`` and this field in the tower of function fields.
INPUT:
- ``base`` -- a function field, either this field or a field from which this field has been created as an extension
OUTPUT:
A list of fields. The first entry is ``base``, the last entry is this field.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: K._intermediate_fields(K) [Rational function field in x over Rational Field]
sage: R.<y> = K[] sage: L.<y> = K.extension(y^2-x) sage: L._intermediate_fields(K) [Function field in y defined by y^2 - x, Rational function field in x over Rational Field]
sage: R.<z> = L[] sage: M.<z> = L.extension(z^2-y) sage: M._intermediate_fields(L) [Function field in z defined by z^2 - y, Function field in y defined by y^2 - x] sage: M._intermediate_fields(K) [Function field in z defined by z^2 - y, Function field in y defined by y^2 - x, Rational function field in x over Rational Field]
TESTS::
sage: K._intermediate_fields(M) Traceback (most recent call last): ... ValueError: field has not been constructed as a finite extension of base sage: K._intermediate_fields(QQ) Traceback (most recent call last): ... TypeError: base must be a function field
"""
def rational_function_field(self): r""" Return the rational function field from which this field has been created as an extension.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: K.rational_function_field() Rational function field in x over Rational Field
sage: R.<y> = K[] sage: L.<y> = K.extension(y^2-x) sage: L.rational_function_field() Rational function field in x over Rational Field
sage: R.<z> = L[] sage: M.<z> = L.extension(z^2-y) sage: M.rational_function_field() Rational function field in x over Rational Field """
def valuation(self, prime): r""" Return the discrete valuation on this function field defined by ``prime``.
INPUT:
- ``prime`` -- a place of the function field, a valuation on a subring, or a valuation on another function field together with information for isomorphisms to and from that function field
EXAMPLES:
We create valuations that correspond to finite rational places of a function field::
sage: K.<x> = FunctionField(QQ) sage: v = K.valuation(1); v (x - 1)-adic valuation sage: v(x) 0 sage: v(x - 1) 1
A place can also be specified with an irreducible polynomial::
sage: v = K.valuation(x - 1); v (x - 1)-adic valuation
Similarly, for a finite non-rational place::
sage: v = K.valuation(x^2 + 1); v (x^2 + 1)-adic valuation sage: v(x^2 + 1) 1 sage: v(x) 0
Or for the infinite place::
sage: v = K.valuation(1/x); v Valuation at the infinite place sage: v(x) -1
Instead of specifying a generator of a place, we can define a valuation on a rational function field by giving a discrete valuation on the underlying polynomial ring::
sage: R.<x> = QQ[] sage: w = valuations.GaussValuation(R, valuations.TrivialValuation(QQ)).augmentation(x - 1, 1) sage: v = K.valuation(w); v (x - 1)-adic valuation
Note that this allows us to specify valuations which do not correspond to a place of the function field::
sage: w = valuations.GaussValuation(R, QQ.valuation(2)) sage: v = K.valuation(w); v 2-adic valuation
The same is possible for valuations with `v(1/x) > 0` by passing in an extra pair of parameters, an isomorphism between this function field and an isomorphic function field. That way you can, for example, indicate that the valuation is to be understood as a valuation on `K[1/x]`, i.e., after applying the substitution `x \mapsto 1/x` (here, the inverse map is also `x \mapsto 1/x`)::
sage: w = valuations.GaussValuation(R, QQ.valuation(2)).augmentation(x, 1) sage: w = K.valuation(w) sage: v = K.valuation((w, K.hom([~K.gen()]), K.hom([~K.gen()]))); v Valuation on rational function field induced by [ Gauss valuation induced by 2-adic valuation, v(x) = 1 ] (in Rational function field in x over Rational Field after x |--> 1/x)
Note that classical valuations at finite places or the infinite place are always normalized such that the uniformizing element has valuation 1::
sage: K.<t> = FunctionField(GF(3)) sage: M.<x> = FunctionField(K) sage: v = M.valuation(x^3 - t) sage: v(x^3 - t) 1
However, if such a valuation comes out of a base change of the ground field, this is not the case anymore. In the example below, the unique extension of ``v`` to ``L`` still has valuation 1 on `x^3 - t` but it has valuation ``1/3`` on its uniformizing element `x - w`::
sage: R.<w> = K[] sage: L.<w> = K.extension(w^3 - t) sage: N.<x> = FunctionField(L) sage: w = v.extension(N) # missing factorization, :trac:`16572` Traceback (most recent call last): ... NotImplementedError sage: w(x^3 - t) # not tested 1 sage: w(x - w) # not tested 1/3
There are several ways to create valuations on extensions of rational function fields::
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x); L Function field in y defined by y^2 - x
A place that has a unique extension can just be defined downstairs::
sage: v = L.valuation(x); v (x)-adic valuation
"""
class FunctionField_polymod(FunctionField): """ A function field defined by a univariate polynomial, as an extension of the base field.
EXAMPLES:
We make a function field defined by a degree 5 polynomial over the rational function field over the rational numbers::
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x
We next make a function field over the above nontrivial function field L::
sage: S.<z> = L[] sage: M.<z> = L.extension(z^2 + y*z + y); M Function field in z defined by z^2 + y*z + y sage: 1/z ((x/(-x^4 - 1))*y^4 - 2*x^2/(-x^4 - 1))*z - 1 sage: z * (1/z) 1
We drill down the tower of function fields::
sage: M.base_field() Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x sage: M.base_field().base_field() Rational function field in x over Rational Field sage: M.base_field().base_field().constant_field() Rational Field sage: M.constant_base_field() Rational Field
.. WARNING::
It is not checked if the polynomial used to define this function field is irreducible Hence it is not guaranteed that this object really is a field! This is illustrated below.
::
sage: K.<x>=FunctionField(QQ) sage: R.<y> = K[] sage: L.<y>=K.extension(x^2-y^2) sage: (y-x)*(y+x) 0 sage: 1/(y-x) 1 sage: y-x==0; y+x==0 False False """ def __init__(self, polynomial, names, element_class = FunctionFieldElement_polymod, category=CAT): """ Create a function field defined as an extension of another function field by adjoining a root of a univariate polynomial.
INPUT:
- ``polynomial`` -- a univariate polynomial over a function field - ``names`` -- variable names (as a tuple of length 1 or string) - ``category`` -- a category (defaults to category of function fields)
EXAMPLES:
We create an extension of a function field::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L = K.extension(y^5 - x^3 - 3*x + x*y); L Function field in y defined by y^5 + x*y - x^3 - 3*x
Note the type::
sage: type(L) <class 'sage.rings.function_field.function_field.FunctionField_polymod_with_category'>
We can set the variable name, which doesn't have to be y::
sage: L.<w> = K.extension(y^5 - x^3 - 3*x + x*y); L Function field in w defined by w^5 + x*w - x^3 - 3*x
TESTS:
Test that :trac:`17033` is fixed::
sage: K.<t> = FunctionField(QQ) sage: R.<x> = QQ[] sage: M.<z> = K.extension(x^7-x-t) sage: M(x) z sage: M('z') z sage: M('x') Traceback (most recent call last): ... TypeError: unable to evaluate 'x' in Fraction Field of Univariate Polynomial Ring in t over Rational Field """ raise TypeError("polynomial must be univariate a polynomial") names = (polynomial.variable_name(), ) raise ValueError("polynomial must have positive degree") raise TypeError("polynomial must be over a FunctionField")
def __reduce__(self): """ Returns the arguments which were used to create this instance. The rationale for this is explained in the documentation of ``UniqueRepresentation``.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L = K.extension(y^2 - x) sage: clazz,args = L.__reduce__() sage: clazz(*args) Function field in y defined by y^2 - x """
def __hash__(self): """ Return hash of this function field.
The hash value is equal to the hash of the defining polynomial.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L = K.extension(y^5 - x^3 - 3*x + x*y) sage: hash(L) == hash(L.polynomial()) True
"""
def _to_base_field(self, f): r""" Return ``f`` as an element of the :meth:`base_field`.
INPUT:
- ``f`` -- an element of this function field which lies in the base field.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: L._to_base_field(L(x)) x sage: L._to_base_field(y) Traceback (most recent call last): ... ValueError: y is not an element of the base field
TESTS:
Verify that :trac:`21872` has been resolved::
sage: R.<z> = L[] sage: M.<z> = L.extension(z^2 - y)
sage: M(1) in QQ True sage: M(y) in L True sage: M(x) in K True sage: z in K False
"""
def _to_constant_base_field(self, f): r""" Return ``f`` as an element of the :meth:`constant_base_field`.
INPUT:
- ``f`` -- an element of this rational function field which is a constant
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: L._to_constant_base_field(L(1)) 1 sage: L._to_constant_base_field(y) Traceback (most recent call last): ... ValueError: y is not an element of the base field
TESTS:
Verify that :trac:`21872` has been resolved::
sage: L(1) in QQ True sage: y in QQ False
"""
def monic_integral_model(self, names=None): """ Return a function field isomorphic to this field but which is an extension of a rational function field with defining polynomial that is monic and integral over the constant base field.
INPUT:
- ``names`` -- a string or a tuple of up to two strings (default: ``None``), the name of the generator of the field, and the name of the generator of the underlying rational function field (if a tuple); if not given, then the names are chosen automatically.
OUTPUT:
A triple ``(F,f,t)`` where ``F`` is a function field, ``f`` is an isomorphism from ``F`` to this field, and ``t`` is the inverse of ``f``.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(x^2*y^5 - 1/x); L Function field in y defined by x^2*y^5 - 1/x sage: A, from_A, to_A = L.monic_integral_model('z') sage: A Function field in z defined by z^5 - x^12 sage: from_A Function Field morphism: From: Function field in z defined by z^5 - x^12 To: Function field in y defined by x^2*y^5 - 1/x Defn: z |--> x^3*y x |--> x sage: to_A Function Field morphism: From: Function field in y defined by x^2*y^5 - 1/x To: Function field in z defined by z^5 - x^12 Defn: y |--> 1/x^3*z x |--> x sage: to_A(y) 1/x^3*z sage: from_A(to_A(y)) y sage: from_A(to_A(1/y)) x^3*y^4 sage: from_A(to_A(1/y)) == 1/y True
This also works for towers of function fields::
sage: R.<z> = L[] sage: M.<z> = L.extension(z^2*y - 1/x) sage: M.monic_integral_model() (Function field in z_ defined by z_^10 - x^18, Function Field morphism: From: Function field in z_ defined by z_^10 - x^18 To: Function field in z defined by y*z^2 - 1/x Defn: z_ |--> x^2*z x |--> x, Function Field morphism: From: Function field in z defined by y*z^2 - 1/x To: Function field in z_ defined by z_^10 - x^18 Defn: z |--> 1/x^2*z_ y |--> 1/x^15*z_^8 x |--> x)
TESTS:
If the field is already a monic integral extension, then it is returned unchanged::
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2-x) sage: L.monic_integral_model() (Function field in y defined by y^2 - x, Function Field endomorphism of Function field in y defined by y^2 - x Defn: y |--> y x |--> x, Function Field endomorphism of Function field in y defined by y^2 - x Defn: y |--> y x |--> x)
unless ``names`` does not match with the current names::
sage: L.monic_integral_model(names=('yy','xx')) (Function field in yy defined by yy^2 - xx, Function Field morphism: From: Function field in yy defined by yy^2 - xx To: Function field in y defined by y^2 - x Defn: yy |--> y xx |--> x, Function Field morphism: From: Function field in y defined by y^2 - x To: Function field in yy defined by yy^2 - xx Defn: y |--> yy x |--> xx)
""" raise ValueErorr("names must contain at most 2 entries")
else: # self is already monic and integral else:
def _make_monic_integral(self, f): r""" Let y be a root of ``f``. This function returns a monic integral polynomial g and an element d of the base field such that g(y*d)=0.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[]; sage: L.<y> = K.extension(x^2*y^5 - 1/x) sage: g, d = L._make_monic_integral(L.polynomial()); g,d (y^5 - x^12, x^3) sage: (y*d).is_integral() True sage: g.is_monic() True sage: g(y*d) 0 """ raise NotImplementedError
# make f monic
# find lcm of denominators # would be good to replace this by minimal... else:
def constant_field(self): """ Return the algebraic closure of the constant field of the base field in this function field.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.constant_field() Traceback (most recent call last): ... NotImplementedError """
def constant_base_field(self): """ Return the constant field of the base rational function field.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x sage: L.constant_base_field() Rational Field sage: S.<z> = L[] sage: M.<z> = L.extension(z^2 - y) sage: M.constant_base_field() Rational Field """
def degree(self, base=None): """ Return the degree of this function field over the function field ``base``.
INPUT:
- ``base`` -- a function field (default: ``None``), a function field from which this field has been constructed as a finite extension.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x sage: L.degree() 5 sage: L.degree(L) 1
sage: R.<z> = L[] sage: M.<z> = L.extension(z^2 - y) sage: M.degree(L) 2 sage: M.degree(K) 10
TESTS::
sage: L.degree(M) Traceback (most recent call last): ... ValueError: base must be None or the rational function field
"""
def _repr_(self): """ Return string representation of this function field.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L._repr_() 'Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x' """
def base_field(self): """ Return the base field of this function field. This function field is presented as L = K[y]/(f(y)), and the base field is by definition the field K.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.base_field() Rational function field in x over Rational Field """
def random_element(self, *args, **kwds): """ Create a random element of this function field. Parameters are passed onto the random_element method of the base_field.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^2 - (x^2 + x)) sage: L.random_element() # random ((x^2 - x + 2/3)/(x^2 + 1/3*x - 1))*y^2 + ((-1/4*x^2 + 1/2*x - 1)/(-5/2*x + 2/3))*y + (-1/2*x^2 - 4)/(-12*x^2 + 1/2*x - 1/95) """
def polynomial(self): """ Return the univariate polynomial that defines this function field, i.e., the polynomial f(y) so that this function field is of the form K[y]/(f(y)).
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.polynomial() y^5 - 2*x*y + (-x^4 - 1)/x """
def is_separable(self): r""" Return whether the defining polynomial of the function field is separable, i.e., whether the gcd of the defining polynomial and its derivative is constant.
EXAMPLES::
sage: K.<x> = FunctionField(GF(5)); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.is_separable() True
sage: K.<x> = FunctionField(GF(5)); R.<y> = K[] sage: L.<y> = K.extension(y^5 - 1) sage: L.is_separable() False
"""
def polynomial_ring(self): """ Return the polynomial ring used to represent elements of this function field. If we view this function field as being presented as K[y]/(f(y)), then this function returns the ring K[y].
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.polynomial_ring() Univariate Polynomial Ring in y over Rational function field in x over Rational Field """
def vector_space(self, base=None): """ Return a vector space `V` and isomorphisms from this field to `V` and from `V` to this field.
This function allows us to identify the elements of this field with elements of a vector space over the base field, which is useful for representation and arithmetic with orders, ideals, etc.
INPUT:
- ``base`` -- a function field (default: ``None``), the returned vector space is over ``base`` which defaults to the base field of this function field.
OUTPUT:
- ``V`` -- a vector space over base field - ``from_V`` -- an isomorphism from V to this field - ``to_V`` -- an isomorphism from this field to V
EXAMPLES:
We define a function field::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x
We get the vector spaces, and maps back and forth::
sage: V, from_V, to_V = L.vector_space() sage: V Vector space of dimension 5 over Rational function field in x over Rational Field sage: from_V Isomorphism morphism: From: Vector space of dimension 5 over Rational function field in x over Rational Field To: Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x sage: to_V Isomorphism morphism: From: Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x To: Vector space of dimension 5 over Rational function field in x over Rational Field
We convert an element of the vector space back to the function field::
sage: from_V(V.1) y
We define an interesting element of the function field::
sage: a = 1/L.0; a (-x/(-x^4 - 1))*y^4 + 2*x^2/(-x^4 - 1)
We convert it to the vector space, and get a vector over the base field::
sage: to_V(a) (2*x^2/(-x^4 - 1), 0, 0, 0, -x/(-x^4 - 1))
We convert to and back, and get the same element::
sage: from_V(to_V(a)) == a True
In the other direction::
sage: v = x*V.0 + (1/x)*V.1 sage: to_V(from_V(v)) == v True
And we show how it works over an extension of an extension field::
sage: R2.<z> = L[]; M.<z> = L.extension(z^2 -y) sage: M.vector_space() (Vector space of dimension 2 over Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x, Isomorphism morphism: From: Vector space of dimension 2 over Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x To: Function field in z defined by z^2 - y, Isomorphism morphism: From: Function field in z defined by z^2 - y To: Vector space of dimension 2 over Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x)
We can also get the vector space of ``M`` over ``K``::
sage: M.vector_space(K) (Vector space of dimension 10 over Rational function field in x over Rational Field, Isomorphism morphism: From: Vector space of dimension 10 over Rational function field in x over Rational Field To: Function field in z defined by z^2 - y, Isomorphism morphism: From: Function field in z defined by z^2 - y To: Vector space of dimension 10 over Rational function field in x over Rational Field)
"""
def maximal_order(self): """ Return the maximal_order of self. If we view self as L = K[y]/(f(y)), then this is the ring of elements of L that are integral over K.
EXAMPLES:
This is not yet implemented...::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.maximal_order() Traceback (most recent call last): ... NotImplementedError """
def _element_constructor_(self, x): r""" Make ``x`` into an element of this function field, possibly not canonically.
INPUT:
- ``x`` -- the element
OUTPUT:
``x``, as an element of this function field
TESTS::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L._element_constructor_(L.polynomial_ring().gen()) y """
def gen(self, n=0): """ Return the ``n``-th generator of this function field. By default ``n`` is 0; any other value of ``n`` leads to an error. The generator is the class of y, if we view self as being presented as K[y]/(f(y)).
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.gen() y sage: L.gen(1) Traceback (most recent call last): ... IndexError: Only one generator. """
def ngens(self): """ Return the number of generators of this function field over its base field. This is by definition 1.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.ngens() 1 """
def equation_order(self): """ If we view self as being presented as K[y]/(f(y)), then this function returns the order generated by the class of y. If f is not monic, then :meth:`_make_monic_integral` is called, and instead we get the order generated by some integral multiple of a root of f.
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: O = L.equation_order() sage: O.basis() (1, x*y, x^2*y^2, x^3*y^3, x^4*y^4)
We try an example, in which the defining polynomial is not monic and is not integral::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(x^2*y^5 - 1/x); L Function field in y defined by x^2*y^5 - 1/x sage: O = L.equation_order() sage: O.basis() (1, x^3*y, x^6*y^2, x^9*y^3, x^12*y^4) """
def hom(self, im_gens, base_morphism=None): """ Create a homomorphism from self to another function field.
INPUT:
- ``im_gens`` -- a list of images of the generators of self and of successive base rings.
- ``base_morphism`` -- (default: None) a homomorphism of the base ring, after the im_gens are used. Thus if im_gens has length 2, then base_morphism should be a morphism from self.base_ring().base_ring().
EXAMPLES:
We create a rational function field, and a quadratic extension of it::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^2 - x^3 - 1)
We make the field automorphism that sends y to -y::
sage: f = L.hom(-y); f Function Field endomorphism of Function field in y defined by y^2 - x^3 - 1 Defn: y |--> -y
Evaluation works::
sage: f(y*x - 1/x) -x*y - 1/x
We try to define an invalid morphism::
sage: f = L.hom(y+1) Traceback (most recent call last): ... ValueError: invalid morphism
We make a morphism of the base rational function field::
sage: phi = K.hom(x+1); phi Function Field endomorphism of Rational function field in x over Rational Field Defn: x |--> x + 1 sage: phi(x^3 - 3) x^3 + 3*x^2 + 3*x - 2 sage: (x+1)^3-3 x^3 + 3*x^2 + 3*x - 2
We make a morphism by specifying where the generators and the base generators go::
sage: L.hom([-y, x]) Function Field endomorphism of Function field in y defined by y^2 - x^3 - 1 Defn: y |--> -y x |--> x
You can also specify a morphism on the base::
sage: R1.<r> = K[] sage: L1.<r> = K.extension(r^2 - (x+1)^3 - 1) sage: L.hom(r, base_morphism=phi) Function Field morphism: From: Function field in y defined by y^2 - x^3 - 1 To: Function field in r defined by r^2 - x^3 - 3*x^2 - 3*x - 2 Defn: y |--> r x |--> x + 1
We make another extension of a rational function field::
sage: K2.<t> = FunctionField(QQ); R2.<w> = K2[] sage: L2.<w> = K2.extension((4*w)^2 - (t+1)^3 - 1)
We define a morphism, by giving the images of generators::
sage: f = L.hom([4*w, t+1]); f Function Field morphism: From: Function field in y defined by y^2 - x^3 - 1 To: Function field in w defined by 16*w^2 - t^3 - 3*t^2 - 3*t - 2 Defn: y |--> 4*w x |--> t + 1
Evaluation works, as expected::
sage: f(y+x) 4*w + t + 1 sage: f(x*y + x/(x^2+1)) (4*t + 4)*w + (t + 1)/(t^2 + 2*t + 2)
We make another extension of a rational function field::
sage: K3.<yy> = FunctionField(QQ); R3.<xx> = K3[] sage: L3.<xx> = K3.extension(yy^2 - xx^3 - 1)
This is the function field L with the generators exchanged. We define a morphism to L::
sage: g = L3.hom([x,y]); g Function Field morphism: From: Function field in xx defined by -xx^3 + yy^2 - 1 To: Function field in y defined by y^2 - x^3 - 1 Defn: xx |--> x yy |--> y
""" raise ValueError("no images specified")
# the codomain of this morphism is the field containing all the im_gens
@cached_method def genus(self): """ Return the genus of this function field For now, the genus is computed using singular
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^3 - (x^3 + 2*x*y + 1/x)) sage: L.genus() 3 """ # unfortunately singular can not compute the genus with the # polynomial_ring()._singular_ object because genus method # only accepts a ring of transcendental degree 2 over a prime # field not a ring of transcendental degree 1 over a rational # function field of one variable
#Making the auxiliary ring which only has polynomials with integral coefficients.
else: raise NotImplementedError("Computation of genus over this rational function field not implemented yet")
@cached_method def derivation(self): r""" Return a derivation of the function field over the constant base field.
A derivation on `R` is a map `R\to R` satisfying `D(\alpha+\beta)=D(\alpha)+D(\beta)` and `D(\alpha\beta)=\beta D(\alpha)+\alpha D(\beta)` for all `\alpha, \beta \in R`. For a function field which is a finite extension of `K(x)` with `K` perfect, the derivations form a one-dimensional `K`-vector space generated by the derivation returned by this method.
OUTPUT:
- a derivation of the function field
EXAMPLES::
sage: K.<x> = FunctionField(GF(3)) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: d = L.derivation(); d Derivation map: From: Function field in y defined by y^2 + 2*x To: Function field in y defined by y^2 + 2*x Defn: y |--> 2/x*y sage: d(x) 1 sage: d(x^3) 0 sage: d(x*y) 0 sage: d(y) 2/x*y
Derivations are linear and satisfy Leibniz's law::
sage: d(x+y) == d(x) + d(y) True sage: d(x*y) == x*d(y) + y*d(x) True
If the field is a separable extension of the base field, the derivation extending a derivation of the base function field is uniquely determined. Proposition 11 of [GT1996]_ describes how to compute the extension. We apply the formula described there to the generator of the space of derivations on the base field.
The general inseparable case is not implemented yet (see :trac:`16562`, :trac:`16564`.)` """ else: raise NotImplementedError("construction of separable models not implemented")
def _simple_model(self, name='v'): r""" Return a finite extension `N/K(x)` isomorphic to the tower of extensions `M/L/K(x)` with `K` perfect.
Helper method for :meth:`simple_model`.
INPUT:
- ``name`` -- a string, the name of the generator of `N`
ALGORITHM:
Since `K` is perfect, the extension `M/K(x)` is simple, i.e., generated by a single element [BM1940]_. Therefore, there are only finitely many intermediate fields (Exercise 3.6.7 in [Bo2009]_). Let `a` be a generator of `M/L` and let `b` be a generator of `L/K(x)`. For some `i` the field `N_i=K(x)(a+x^ib)` is isomorphic to `M` and so it is enough to test for all terms of the form `a+x^ib` whether they generate a field of the right degree. Indeed, suppose for contradiction that for all `i` we had `N_i\neq M`. Then `N_i=N_j` for some `i,j`. Thus `(a+x^ib)-(a+x^jb)=b(x^i-x^j)\in N_j` and so `b\in N_j`. Similarly, `a+x^ib-x^{i-j}(a+x^jb)=a(1+x^{i-j})\in N_j` and so `a\in N_j`. Therefore, `N_j=M`.
TESTS::
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2-x) sage: R.<z> = L[] sage: M.<z> = L.extension(z^2-y) sage: M._simple_model() (Function field in v defined by v^4 - x, Function Field morphism: From: Function field in v defined by v^4 - x To: Function field in z defined by z^2 - y Defn: v |--> z, Function Field morphism: From: Function field in z defined by z^2 - y To: Function field in v defined by v^4 - x Defn: z |--> v y |--> v^2)
Check that this also works for inseparable extensions::
sage: K.<x> = FunctionField(GF(2)) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2-x) sage: R.<z> = L[] sage: M.<z> = L.extension(z^2-y) sage: M._simple_model() (Function field in v defined by v^4 + x, Function Field morphism: From: Function field in v defined by v^4 + x To: Function field in z defined by z^2 + y Defn: v |--> z, Function Field morphism: From: Function field in z defined by z^2 + y To: Function field in v defined by v^4 + x Defn: z |--> v y |--> v^2)
An example where the generator of the last extension does not generate the extension of the rational function field::
sage: K.<x> = FunctionField(GF(2)) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2-x) sage: R.<z> = L[] sage: M.<z> = L.extension(z^3-1) sage: M._simple_model() (Function field in v defined by v^6 + x*v^4 + x^2*v^2 + x^3 + 1, Function Field morphism: From: Function field in v defined by v^6 + x*v^4 + x^2*v^2 + x^3 + 1 To: Function field in z defined by z^3 + 1 Defn: v |--> z + y, Function Field morphism: From: Function field in z defined by z^3 + 1 To: Function field in v defined by v^6 + x*v^4 + x^2*v^2 + x^3 + 1 Defn: z |--> v^4 + x^2 y |--> v^4 + v + x^2)
"""
raise NotImplementedError("simple_model() only implemented over perfect constant fields")
# using a+x^i*b tends to lead to huge powers of x in the minimal # polynomial of the resulting field; it is better to try terms of # the form a+i*b first (but in characteristic p>0 there are only # finitely many of these) # We systematically try elements of the form a+b*factor*x^exponent factor = self.constant_base_field().one() exponent += 1
# the morphism N -> M, v |-> v
# the morphism M -> N, b |-> M_b, a |-> M_a # the power basis of v over K
@cached_method def simple_model(self, name=None): """ Return a function field isomorphic to this field which is a simple extension of a rational function field.
INPUT:
- ``name`` -- a string (default: ``None``), the name of generator of the simple extension. If ``None``, then the name of the generator will be the same as the name of the generator of this function field.
OUTPUT:
A triple ``(F,f,t)`` where ``F`` is a field isomorphic to this field, ``f`` is an isomorphism from ``F`` to this function field and ``t`` is the inverse of ``f``.
EXAMPLES:
A tower of four function fields::
sage: K.<x> = FunctionField(QQ); R.<z> = K[] sage: L.<z> = K.extension(z^2-x); R.<u> = L[] sage: M.<u> = L.extension(u^2-z); R.<v> = M[] sage: N.<v> = M.extension(v^2-u)
The fields N and M as simple extensions of K::
sage: N.simple_model() (Function field in v defined by v^8 - x, Function Field morphism: From: Function field in v defined by v^8 - x To: Function field in v defined by v^2 - u Defn: v |--> v, Function Field morphism: From: Function field in v defined by v^2 - u To: Function field in v defined by v^8 - x Defn: v |--> v u |--> v^2 z |--> v^4 x |--> x) sage: M.simple_model() (Function field in u defined by u^4 - x, Function Field morphism: From: Function field in u defined by u^4 - x To: Function field in u defined by u^2 - z Defn: u |--> u, Function Field morphism: From: Function field in u defined by u^2 - z To: Function field in u defined by u^4 - x Defn: u |--> u z |--> u^2 x |--> x)
An optional parameter ``name`` can be used to set the name of the generator of the simple extension::
sage: M.simple_model(name='t') (Function field in t defined by t^4 - x, Function Field morphism: From: Function field in t defined by t^4 - x To: Function field in u defined by u^2 - z Defn: t |--> u, Function Field morphism: From: Function field in u defined by u^2 - z To: Function field in t defined by t^4 - x Defn: u |--> t z |--> t^2 x |--> x)
An example with higher degrees::
sage: K.<x> = FunctionField(GF(3)); R.<y> = K[] sage: L.<y> = K.extension(y^5-x); R.<z> = L[] sage: M.<z> = L.extension(z^3-x) sage: M.simple_model() (Function field in z defined by z^15 + x*z^12 + x^2*z^9 + 2*x^3*z^6 + 2*x^4*z^3 + 2*x^5 + 2*x^3, Function Field morphism: From: Function field in z defined by z^15 + x*z^12 + x^2*z^9 + 2*x^3*z^6 + 2*x^4*z^3 + 2*x^5 + 2*x^3 To: Function field in z defined by z^3 + 2*x Defn: z |--> z + y, Function Field morphism: From: Function field in z defined by z^3 + 2*x To: Function field in z defined by z^15 + x*z^12 + x^2*z^9 + 2*x^3*z^6 + 2*x^4*z^3 + 2*x^5 + 2*x^3 Defn: z |--> 2/x*z^6 + 2*z^3 + z + 2*x y |--> 1/x*z^6 + z^3 + x x |--> x)
This also works for inseparable extensions::
sage: K.<x> = FunctionField(GF(2)); R.<y> = K[] sage: L.<y> = K.extension(y^2-x); R.<z> = L[] sage: M.<z> = L.extension(z^2-y) sage: M.simple_model() (Function field in z defined by z^4 + x, Function Field morphism: From: Function field in z defined by z^4 + x To: Function field in z defined by z^2 + y Defn: z |--> z, Function Field morphism: From: Function field in z defined by z^2 + y To: Function field in z defined by z^4 + x Defn: z |--> z y |--> z^2 x |--> x) """
# the extension is simple already else: ret = self.base_field().extension(self.polynomial(), names=(name,)) f = ret.hom(self.gen()) t = self.hom(ret.gen()) return ret, f, t else: # recursively collapse the tower of fields for k in base._intermediate_fields(base.rational_function_field())]
# now collapse self_/base_/K(x) for k in self._intermediate_fields(self.rational_function_field())]
@cached_method def primitive_element(self): r""" Return a primitive element over the underlying rational function field.
If this is a finite extension of a rational function field `K(x)` with `K` perfect, then this is a simple extension of `K(x)`, i.e., there is a primitive element `y` which generates this field over `K(x)`. This method returns such an element `y`.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2-x) sage: R.<z> = L[] sage: M.<z> = L.extension(z^2-y) sage: R.<z> = L[] sage: N.<u> = L.extension(z^2-x-1) sage: N.primitive_element() u + y sage: M.primitive_element() z sage: L.primitive_element() y
This also works for inseparable extensions::
sage: K.<x> = FunctionField(GF(2)) sage: R.<Y> = K[] sage: L.<y> = K.extension(Y^2-x) sage: R.<Z> = L[] sage: M.<z> = L.extension(Z^2-y) sage: M.primitive_element() z """
def change_variable_name(self, name): r""" Return a field isomorphic to this field with variable(s) ``name``.
INPUT:
- ``name`` -- a string or a tuple consisting of a strings, the names of the new variables starting with a generator of this field and going down to the rational function field.
OUTPUT:
A triple ``F,f,t`` where ``F`` is a function field, ``f`` is an isomorphism from ``F`` to this field, and ``t`` is the inverse of ``f``.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: R.<z> = L[] sage: M.<z> = L.extension(z^2 - y)
sage: M.change_variable_name('zz') (Function field in zz defined by zz^2 - y, Function Field morphism: From: Function field in zz defined by zz^2 - y To: Function field in z defined by z^2 - y Defn: zz |--> z y |--> y x |--> x, Function Field morphism: From: Function field in z defined by z^2 - y To: Function field in zz defined by zz^2 - y Defn: z |--> zz y |--> y x |--> x) sage: M.change_variable_name(('zz','yy')) (Function field in zz defined by zz^2 - yy, Function Field morphism: From: Function field in zz defined by zz^2 - yy To: Function field in z defined by z^2 - y Defn: zz |--> z yy |--> y x |--> x, Function Field morphism: From: Function field in z defined by z^2 - y To: Function field in zz defined by zz^2 - yy Defn: z |--> zz y |--> yy x |--> x) sage: M.change_variable_name(('zz','yy','xx')) (Function field in zz defined by zz^2 - yy, Function Field morphism: From: Function field in zz defined by zz^2 - yy To: Function field in z defined by z^2 - y Defn: zz |--> z yy |--> y xx |--> x, Function Field morphism: From: Function field in z defined by z^2 - y To: Function field in zz defined by zz^2 - yy Defn: z |--> zz y |--> yy x |--> xx)
""" raise ValueError("name must contain at least one string") else:
def is_RationalFunctionField(x): """ Return ``True`` if ``x`` is of rational function field type.
EXAMPLES::
sage: from sage.rings.function_field.function_field import is_RationalFunctionField sage: is_RationalFunctionField(QQ) False sage: is_RationalFunctionField(FunctionField(QQ,'t')) True """
class RationalFunctionField(FunctionField): """ A rational function field K(t) in one variable, over an arbitrary base field.
EXAMPLES::
sage: K.<t> = FunctionField(GF(3)); K Rational function field in t over Finite Field of size 3 sage: K.gen() t sage: 1/t + t^3 + 5 (t^4 + 2*t + 1)/t
There are various ways to get at the underlying fields and rings associated to a rational function field::
sage: K.<t> = FunctionField(GF(7)) sage: K.base_field() Rational function field in t over Finite Field of size 7 sage: K.field() Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 sage: K.constant_field() Finite Field of size 7 sage: K.maximal_order() Maximal order in Rational function field in t over Finite Field of size 7
We define a morphism::
sage: K.<t> = FunctionField(QQ) sage: L = FunctionField(QQ, 'tbar') # give variable name as second input sage: K.hom(L.gen()) Function Field morphism: From: Rational function field in t over Rational Field To: Rational function field in tbar over Rational Field Defn: t |--> tbar """ def __init__(self, constant_field, names, element_class = FunctionFieldElement_rational, category=CAT): """ Create a rational function field in one variable.
INPUT:
- ``constant_field`` -- an arbitrary field - ``names`` -- a string or tuple of length 1 - ``category`` -- default: FunctionFields()
EXAMPLES::
sage: K.<t> = FunctionField(CC); K Rational function field in t over Complex Field with 53 bits of precision sage: K.category() Category of function fields sage: FunctionField(QQ[I], 'alpha') Rational function field in alpha over Number Field in I with defining polynomial x^2 + 1
Must be over a field::
sage: FunctionField(ZZ, 't') Traceback (most recent call last): ... TypeError: constant_field must be a field """ raise ValueError("variable name must be specified") names = (names, )
def __reduce__(self): """ Returns the arguments which were used to create this instance. The rationale for this is explained in the documentation of ``UniqueRepresentation``.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: clazz,args = K.__reduce__() sage: clazz(*args) Rational function field in x over Rational Field """
def __hash__(self): """ Return hash of this function field.
The hash is formed from the constant field and the variable names.
EXAMPLES::
sage: K.<t> = FunctionField(QQ) sage: hash(K) == hash((K.constant_base_field(), K.variable_names())) True
"""
def _repr_(self): """ Return string representation of this function field.
EXAMPLES::
sage: K.<t> = FunctionField(QQ) sage: K._repr_() 'Rational function field in t over Rational Field' """ self.variable_name(), self._constant_field)
def _element_constructor_(self, x): r""" Coerce ``x`` into an element of this function field, possibly not canonically.
INPUT:
- ``x`` -- the element
OUTPUT:
``x``, as an element of this function field
EXAMPLES::
sage: K.<t> = FunctionField(QQ) sage: a = K._element_constructor_(K.maximal_order().gen()); a t sage: a.parent() Rational function field in t over Rational Field
TESTS:
Conversion of a string::
sage: K('t') t sage: K('1/t') 1/t
Conversion of a constant polynomial over the function field::
sage: K(K.polynomial_ring().one()) 1
Some indirect test of conversion::
sage: S.<x, y> = K[] sage: I = S*[x^2 - y^2, y-t] sage: I.groebner_basis() [x^2 - t^2, y - t]
""" return x[0]
def _to_constant_base_field(self, f): r""" Return ``f`` as an element of the :meth:`constant_base_field`.
INPUT:
- ``f`` -- an element of this rational function field which is a constant of the underlying rational function field.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: K._to_constant_base_field(K(1)) 1 sage: K._to_constant_base_field(K(x)) Traceback (most recent call last): ... ValueError: only constants can be converted into the constant base field but x is not a constant
TESTS:
Verify that :trac:`21872` has been resolved::
sage: K(1) in QQ True sage: x in QQ False
""" # When K is not exact, f.denominator() might not be an exact 1, so # we need to divide explicitly to get the correct precision
def _to_polynomial(self, f): """ If ``f`` is integral, return it as a polynomial.
INPUT:
- ``f`` -- an element of this rational function field whose denominator is a constant.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: K._ring(x) # indirect doctest x """
def _to_bivariate_polynomial(self, f): """ Convert ``f`` from a univariate polynomial over the rational function field into a bivariate polynomial and a denominator.
INPUT:
- ``f`` -- a univariate polynomial over self.
OUTPUT:
- bivariate polynomial, denominator
EXAMPLES::
sage: R.<t> = FunctionField(GF(7)) sage: S.<X> = R[] sage: f = (1/t)*(X^4 - 1/t^2)*(X^3 - t^3) sage: R._to_bivariate_polynomial(f) (X^7*t^2 - X^4*t^5 - X^3 + t^3, t^3) """
def _factor_univariate_polynomial(self, f, proof=True): """ Factor the univariate polynomial ``f`` over self.
EXAMPLES::
We do a factorization over the function field over the rationals::
sage: R.<t> = FunctionField(QQ) sage: S.<X> = R[] sage: f = (1/t)*(X^4 - 1/t^2)*(X^3 - t^3) sage: f.factor() # indirect doctest (1/t) * (X - t) * (X^2 - 1/t) * (X^2 + 1/t) * (X^2 + t*X + t^2) sage: f.factor().prod() == f True
We do a factorization over a finite prime field::
sage: R.<t> = FunctionField(GF(7)) sage: S.<X> = R[] sage: f = (1/t)*(X^4 - 1/t^2)*(X^3 - t^3) sage: f.factor() (1/t) * (X + 3*t) * (X + 5*t) * (X + 6*t) * (X^2 + 1/t) * (X^2 + 6/t) sage: f.factor().prod() == f True
Factoring over a function field over a non-prime finite field::
sage: k.<a> = GF(9) sage: R.<t> = FunctionField(k) sage: S.<X> = R[] sage: f = (1/t)*(X^3 - a*t^3) sage: f.factor() (1/t) * (X + (a + 2)*t)^3 sage: f.factor().prod() == f True
We check that ``proof`` parameter is passed to the underlying polynomial (see :trac:`24510`). However, factoring over a function field over a tower of finite fields does not work yet (see :trac:`24533`)::
sage: k = GF(4) sage: k.<a> = GF(4) sage: R.<b> = k[] sage: l.<b> = k.extension(a^2 + a + b) sage: K.<x> = FunctionField(l) sage: R.<t> = K[] sage: F = t*x sage: F.factor(proof=False) Traceback (most recent call last): ... TypeError: no conversion of this ring to a Singular ring defined
""" # the variables of the bivariate polynomial must be distinct # replace x with xx to make the variable names distinct
# undo any variable substitution that we introduced for the bivariate polynomial
@cached_method def polynomial_ring(self, var='x'): """ Return a polynomial ring in one variable over this rational function field.
INPUT:
- ``var`` -- a string (default: 'x')
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: K.polynomial_ring() Univariate Polynomial Ring in x over Rational function field in x over Rational Field sage: K.polynomial_ring('T') Univariate Polynomial Ring in T over Rational function field in x over Rational Field """
@cached_method def vector_space(self): """ Return a vector space V and isomorphisms self --> V and V --> self.
OUTPUT:
- ``V`` -- a vector space over the rational numbers - ``from_V`` -- an isomorphism from V to self - ``to_V`` -- an isomorphism from self to V
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: K.vector_space() (Vector space of dimension 1 over Rational function field in x over Rational Field, Isomorphism morphism: From: Vector space of dimension 1 over Rational function field in x over Rational Field To: Rational function field in x over Rational Field, Isomorphism morphism: From: Rational function field in x over Rational Field To: Vector space of dimension 1 over Rational function field in x over Rational Field) """ V = self.base_field()**1 from .maps import MapVectorSpaceToFunctionField, MapFunctionFieldToVectorSpace from_V = MapVectorSpaceToFunctionField(V, self) to_V = MapFunctionFieldToVectorSpace(self, V) return (V, from_V, to_V)
def random_element(self, *args, **kwds): """ Create a random element of this rational function field.
Parameters are passed to the random_element method of the underlying fraction field.
EXAMPLES::
sage: FunctionField(QQ,'alpha').random_element() # random (-1/2*alpha^2 - 4)/(-12*alpha^2 + 1/2*alpha - 1/95) """
def degree(self, base=None): """ Return the degree over the base field of this rational function field. Since the base field is the rational function field itself, the degree is 1.
INPUT:
- ``base`` -- the base field of the vector space; must be the function field itself (the default)
EXAMPLES::
sage: K.<t> = FunctionField(QQ) sage: K.degree() 1 """
def gen(self, n=0): """ Return the ``n``-th generator of this function field. If ``n`` is not 0, then an IndexError is raised.
EXAMPLES::
sage: K.<t> = FunctionField(QQ); K.gen() t sage: K.gen().parent() Rational function field in t over Rational Field sage: K.gen(1) Traceback (most recent call last): ... IndexError: Only one generator. """
def ngens(self): """ Return the number of generators, which is 1.
EXAMPLES::
sage: K.<t> = FunctionField(QQ) sage: K.ngens() 1 """
def base_field(self): """ Return the base field of this rational function field, which is just this function field itself.
EXAMPLES::
sage: K.<t> = FunctionField(GF(7)) sage: K.base_field() Rational function field in t over Finite Field of size 7 """
def hom(self, im_gens, base_morphism=None): """ Create a homomorphism from self to another ring.
INPUT:
- ``im_gens`` -- exactly one element of some ring. It must be invertible and transcendental over the image of ``base_morphism``; this is not checked. - ``base_morphism`` -- a homomorphism from the base field into the other ring. If ``None``, try to use a coercion map.
OUTPUT:
- a map between function fields
EXAMPLES:
We make a map from a rational function field to itself::
sage: K.<x> = FunctionField(GF(7)) sage: K.hom( (x^4 + 2)/x) Function Field endomorphism of Rational function field in x over Finite Field of size 7 Defn: x |--> (x^4 + 2)/x
We construct a map from a rational function field into a non-rational extension field::
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[] sage: L.<y> = K.extension(y^3 + 6*x^3 + x) sage: f = K.hom(y^2 + y + 2); f Function Field morphism: From: Rational function field in x over Finite Field of size 7 To: Function field in y defined by y^3 + 6*x^3 + x Defn: x |--> y^2 + y + 2 sage: f(x) y^2 + y + 2 sage: f(x^2) 5*y^2 + (x^3 + 6*x + 4)*y + 2*x^3 + 5*x + 4 """ return self.Hom(im_gens).natural_map() raise ValueError("there must be exactly one generator") raise ValueError("You must specify a morphism on the base field")
def field(self): """ Return the underlying field, forgetting the function field structure.
EXAMPLES::
sage: K.<t> = FunctionField(GF(7)) sage: K.field() Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7
.. SEEALSO::
:meth:`sage.rings.fraction_field.FractionField_1poly_field.function_field`
"""
@cached_method def maximal_order(self): """ Return the maximal order of this function field. Since this is a rational function field it is of the form K(t), and the maximal order is by definition K[t].
EXAMPLES::
sage: K.<t> = FunctionField(QQ) sage: K.maximal_order() Maximal order in Rational function field in t over Rational Field sage: K.equation_order() Maximal order in Rational function field in t over Rational Field """
equation_order = maximal_order
def constant_base_field(self): """ Return the field that this rational function field is a transcendental extension of.
EXAMPLES::
sage: K.<t> = FunctionField(QQ) sage: K.constant_field() Rational Field
"""
constant_field = constant_base_field
def genus(self): """ Return the genus of this function field This is always equal 0 for a rational function field
EXAMPLES::
sage: K.<x> = FunctionField(QQ); sage: K.genus() 0 """
def vector_space(self, base=None): """ Return a vector space `V` and isomorphisms from this field to `V` and from `V` to this field.
This function allows us to identify the elements of this field with elements of a one-dimensional vector space over the field itself. This method exists so that all function fields (rational or not) have the same interface.
INPUT:
- ``base`` -- the base field of the vector space; must be the function field itself (the default)
OUTPUT:
- ``V`` -- a vector space over base field - ``from_V`` -- an isomorphism from V to this field - ``to_V`` -- an isomorphism from this field to V
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: K.vector_space() (Vector space of dimension 1 over Rational function field in x over Rational Field, Isomorphism morphism: From: Vector space of dimension 1 over Rational function field in x over Rational Field To: Rational function field in x over Rational Field, Isomorphism morphism: From: Rational function field in x over Rational Field To: Vector space of dimension 1 over Rational function field in x over Rational Field)
TESTS::
sage: K.vector_space() (Vector space of dimension 1 over Rational function field in x over Rational Field, Isomorphism morphism: From: Vector space of dimension 1 over Rational function field in x over Rational Field To: Rational function field in x over Rational Field, Isomorphism morphism: From: Rational function field in x over Rational Field To: Vector space of dimension 1 over Rational function field in x over Rational Field)
""" raise ValueError("base must be the rational function field or None")
def change_variable_name(self, name): r""" Return a field isomorphic to this field with variable ``name``.
INPUT:
- ``name`` -- a string or a tuple consisting of a single string, the name of the new variable
OUTPUT:
A triple ``F,f,t`` where ``F`` is a rational function field, ``f`` is an isomorphism from ``F`` to this field, and ``t`` is the inverse of ``f``.
EXAMPLES::
sage: K.<x> = FunctionField(QQ) sage: L,f,t = K.change_variable_name('y') sage: L,f,t (Rational function field in y over Rational Field, Function Field morphism: From: Rational function field in y over Rational Field To: Rational function field in x over Rational Field Defn: y |--> x, Function Field morphism: From: Rational function field in x over Rational Field To: Rational function field in y over Rational Field Defn: x |--> y) sage: L.change_variable_name('x')[0] is K True
""" raise ValueError("names must be a tuple with a single string") else:
@cached_method def derivation(self): r""" Return a derivation of the rational function field over the constant base field.
OUTPUT:
- a derivation of the rational function field
The derivation maps the generator of the rational function field to 1.
EXAMPLES::
sage: K.<x> = FunctionField(GF(3)) sage: m = K.derivation(); m Derivation map: From: Rational function field in x over Finite Field of size 3 To: Rational function field in x over Finite Field of size 3 sage: m(x) 1
TESTS::
sage: L.<y> = FunctionField(K) sage: L.derivation() Traceback (most recent call last): ... NotImplementedError: not implemented for non-perfect base fields """ |