Coverage for local/lib/python2.7/site-packages/sage/rings/padics/unramified_extension_generic.py : 51%

""" 

Unramified Extension Generic 

This file implements the shared functionality for unramified extensions. 

AUTHORS: 

- David Roe 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2008 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 .padic_extension_generic import pAdicExtensionGeneric 

from .misc import precprint 

from sage.rings.finite_rings.finite_field_constructor import GF 

from sage.misc.cachefunc import cached_method 

class UnramifiedExtensionGeneric(pAdicExtensionGeneric): 

""" 

An unramified extension of Qp or Zp. 

""" 

def __init__(self, poly, prec, print_mode, names, element_class): 

""" 

Initializes self 

INPUT: 

- poly -- Polynomial defining this extension. 

- prec -- The precision cap 

- print_mode -- a dictionary with print options 

- names -- a 4-tuple, (variable_name, residue_name, 

unramified_subextension_variable_name, uniformizer_name) 

- element_class -- the class for elements of this unramified extension. 

EXAMPLES:: 

sage: R.<a> = Zq(27) #indirect doctest 

""" 

#base = poly.base_ring() 

#if base.is_field(): 

# self._PQR = pqr.PolynomialQuotientRing_field(poly.parent(), poly, name = names) 

#else: 

# self._PQR = pqr.PolynomialQuotientRing_domain(poly.parent(), poly, name = names) 

pAdicExtensionGeneric.__init__(self, poly, prec, print_mode, names, element_class) 

self._res_field = GF(self.prime_pow.pow_Integer_Integer(poly.degree()), name = names[1], modulus = poly.change_ring(poly.base_ring().residue_field())) 

def _repr_(self, do_latex = False): 

r""" 

Representation. 

EXAMPLES:: 

sage: R.<a> = Zq(125); R #indirect doctest 

Unramified Extension in a defined by x^3 + 3*x + 3 with capped relative precision 20 over 5-adic Ring 

sage: latex(R) #indirect doctest 

\mathbf{Z}_{5^{3}} 

""" 

if do_latex: 

if self.is_field(): 

return "\\mathbf{Q}_{%s^{%s}}" % (self.prime(), self.degree()) 

else: 

return "\\mathbf{Z}_{%s^{%s}}" % (self.prime(), self.degree()) 

return "Unramified Extension in %s defined by %s %s over %s-adic %s"%(self.variable_name(), self.defining_polynomial(exact=True), precprint(self._prec_type(), self.precision_cap(), self.prime()), self.prime(), "Field" if self.is_field() else "Ring") 

def ramification_index(self, K = None): 

""" 

Returns the ramification index of self over the subring K. 

INPUT: 

- K -- a subring (or subfield) of self. Defaults to the 

base. 

EXAMPLES:: 

sage: R.<a> = Zq(125); R.ramification_index() 

1 

""" 

if K is None: 

return 1 

elif K is self: 

return 1 

else: 

raise NotImplementedError 

def inertia_degree(self, K = None): 

""" 

Returns the inertia degree of self over the subring K. 

INPUT: 

- K -- a subring (or subfield) of self. Defaults to the 

base. 

EXAMPLES:: 

sage: R.<a> = Zq(125); R.inertia_degree() 

3 

""" 

if K is None: 

return self.modulus().degree() 

elif K is self: 

return 1 

else: 

raise NotImplementedError 

#def extension(self, *args, **kwds): 

# raise NotImplementedError 

#def get_extension(self): 

# raise NotImplementedError 

def residue_class_field(self): 

""" 

Returns the residue class field. 

EXAMPLES:: 

sage: R.<a> = Zq(125); R.residue_class_field() 

Finite Field in a0 of size 5^3 

""" 

#should eventually take advantage of finite field 

#\code{extension} or finite field 

#\code{unramified_extension_of_degree} over the automatic 

#coercion base. 

return self._res_field 

def residue_ring(self, n): 

""" 

Return the quotient of the ring of integers by the nth power of its maximal ideal. 

EXAMPLES:: 

sage: R.<a> = Zq(125) 

sage: R.residue_ring(1) 

Finite Field in a0 of size 5^3 

The following requires implementing more general Artinian rings:: 

sage: R.residue_ring(2) 

Traceback (most recent call last): 

... 

NotImplementedError 

""" 

if n == 1: 

return self._res_field 

else: 

raise NotImplementedError 

def discriminant(self, K=None): 

""" 

Returns the discriminant of self over the subring K. 

INPUT: 

- K -- a subring/subfield (defaults to the base ring). 

EXAMPLES:: 

sage: R.<a> = Zq(125) 

sage: R.discriminant() 

Traceback (most recent call last): 

... 

NotImplementedError 

""" 

if K is self: 

return 1 

else: 

raise NotImplementedError 

#def automorphisms(self): 

# raise NotImplementedError 

#def galois_group(self): 

# r""" 

# Returns the Galois group of self's fraction field over Qp. 

# """ 

# ## 

# ## If K is a number field, then K.galois_group() can return 

# ## other variants, i.e. via Pari or KASH. We could consider 

# ## doing this. 

# ## 

# from sage.groups.perm_gps.permgroup import CyclicPermutationGroup 

# return CyclicPermutationGroup(self.modulus().degree()) 

#def is_abelian(self): 

# return True 

def is_galois(self, K=None): 

""" 

Returns True if this extension is Galois. 

Every unramified extension is Galois. 

INPUT: 

- K -- a subring/subfield (defaults to the base ring). 

EXAMPLES:: 

sage: R.<a> = Zq(125); R.is_galois() 

True 

""" 

if K is None or K is self: 

return True 

raise NotImplementedError 

def gen(self, n=0): 

""" 

Returns a generator for this unramified extension. 

This is an element that satisfies the polynomial defining this 

extension. Such an element will reduce to a generator of the 

corresponding residue field extension. 

EXAMPLES:: 

sage: R.<a> = Zq(125); R.gen() 

a + O(5^20) 

""" 

if n != 0: 

raise IndexError("only one generator") 

return self([0,1]) 

@cached_method 

def _frob_gen(self, arithmetic = True): 

""" 

Returns frobenius of the generator for this unramified extension 

EXAMPLES:: 

sage: R.<a> = Zq(9) 

sage: R._frob_gen() 

(2*a + 1) + (2*a + 2)*3 + (2*a + 2)*3^2 + (2*a + 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 + O(3^20) 

""" 

p = self.prime() 

exp = p 

a = self.gen() 

if not arithmetic: 

exp = p**(self.degree()-1) 

approx = (self(a.residue()**exp)).lift_to_precision(self.precision_cap()) #first approximation 

f = self.defining_polynomial() 

g = f.derivative() 

while(f(approx) != 0): #hensel lift frobenius(a) 

approx = approx - f(approx)/g(approx) 

return approx 

def uniformizer_pow(self, n): 

""" 

Returns the nth power of the uniformizer of self (as an element of self). 

EXAMPLES:: 

sage: R.<a> = ZqCR(125) 

sage: R.uniformizer_pow(5) 

5^5 + O(5^25) 

""" 

return self(self.prime_pow(n)) 

def uniformizer(self): 

""" 

Returns a uniformizer for this extension. 

Since this extension is unramified, a uniformizer for the 

ground ring will also be a uniformizer for this extension. 

EXAMPLES:: 

sage: R.<a> = ZqCR(125) 

sage: R.uniformizer() 

5 + O(5^21) 

""" 

return self(self.ground_ring().uniformizer()) 

def _uniformizer_print(self): 

""" 

Returns how the uniformizer is supposed to print. 

EXAMPLES:: 

sage: R.<a> = Zq(125); R._uniformizer_print() 

'5' 

""" 

return self.ground_ring()._uniformizer_print() 

def _unram_print(self): 

""" 

Returns how the generator prints. 

EXAMPLES:: 

sage: R.<a> = Zq(125); R._unram_print() 

'a' 

""" 

return self.variable_name() 

def has_pth_root(self): 

r""" 

Returns whether or not `\ZZ_p` has a primitive `p^{\mbox{th}}` root of unity. 

Since adjoining a `p^{\mbox{th}}` root of unity yields a 

totally ramified extension, self will contain one if and only 

if the ground ring does. 

INPUT: 

- self -- a p-adic ring 

OUTPUT: 

- boolean -- whether self has primitive `p^{\mbox{th}}` 

root of unity. 

EXAMPLES:: 

sage: R.<a> = Zq(1024); R.has_pth_root() 

True 

sage: R.<a> = Zq(17^5); R.has_pth_root() 

False 

""" 

return self.ground_ring().has_pth_root() 

def has_root_of_unity(self, n): 

""" 

Returns whether or not `\ZZ_p` has a primitive `n^{\mbox{th}}` 

root of unity. 

INPUT: 

- self -- a p-adic ring 

- n -- an integer 

OUTPUT: 

- boolean -- whether self has primitive `n^{\mbox{th}}` 

root of unity 

EXAMPLES:: 

sage: R.<a> = Zq(37^8) 

sage: R.has_root_of_unity(144) 

True 

sage: R.has_root_of_unity(89) 

True 

sage: R.has_root_of_unity(11) 

False 

""" 

if (self.prime() == 2): 

return n.divides(2*(self.residue_class_field().order()-1)) 

else: 

return n.divides(self.residue_class_field().order() - 1)