Coverage for local/lib/python2.7/site-packages/sage/rings/padics/eisenstein_extension_generic.py : 40%

""" 

Eisenstein Extension Generic 

This file implements the shared functionality for Eisenstein 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.infinity import infinity 

from sage.misc.latex import latex 

from sage.rings.integer import Integer 

class EisensteinExtensionGeneric(pAdicExtensionGeneric): 

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

""" 

Initializes self. 

EXAMPLES:: 

sage: A = Zp(7,10) 

sage: S.<x> = A[] 

sage: B.<t> = A.ext(x^2+7) #indirect doctest 

""" 

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

#self._precompute() 

def _repr_(self, do_latex = False): 

""" 

Returns a print representation of this extension. 

EXAMPLES:: 

sage: A = Zp(7,10) 

sage: S.<x> = A[] 

sage: B.<t> = A.ext(x^2+7) 

sage: B #indirect doctest 

Eisenstein Extension in t defined by x^2 + 7 with capped relative precision 20 over 7-adic Ring 

""" 

if do_latex: 

return "Eisenstein Extension in %s defined by %s over %s"%(self.latex_name(), latex(self.defining_polynomial(exact=True)), latex(self.ground_ring())) 

else: 

return "Eisenstein 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.variable_name()), self.prime(), "Field" if self.is_field() else "Ring") 

def ramification_index(self, K = None): 

""" 

Returns the ramification index of self over K, or over the 

ground ring if K is None. 

The ramification index is the index of the image of the 

valuation map on K in the image of the valuation map on self 

(both normalized so that the valuation of p is 1). 

INPUT: 

- self -- an Eisenstein extension 

- K -- a subring of self (default None -> self.ground_ring()) 

OUTPUT: 

- The ramification index of the extension self/K 

EXAMPLES:: 

sage: A = Zp(7,10) 

sage: S.<x> = A[] 

sage: B.<t> = A.ext(x^2+7) 

sage: B.ramification_index() 

2 

""" 

if K is None or K is self.ground_ring(): 

return self.modulus().degree() 

elif K is self: 

return 1 

else: 

raise NotImplementedError 

def inertia_degree(self, K = None): 

""" 

Returns the inertia degree of self over K, or the ground ring 

if K is None. 

The inertia degree is the degree of the extension of residue 

fields induced by this extensions. Since Eisenstein 

extensions are totally ramified, this will be 1 for K=None. 

INPUT: 

- self -- an Eisenstein extension 

- K -- a subring of self (default None -> self.ground_ring()) 

OUTPUT: 

- The degree of the induced extensions of residue fields. 

EXAMPLES:: 

sage: A = Zp(7,10) 

sage: S.<x> = A[] 

sage: B.<t> = A.ext(x^2+7) 

sage: B.inertia_degree() 

1 

""" 

if K is None or K is self.ground_ring(): 

return Integer(1) 

elif K is self: 

return Integer(1) 

else: 

raise NotImplementedError 

def inertia_subring(self): 

""" 

Returns the inertia subring. 

Since an Eisenstein extension is totally ramified, this is 

just the ground field. 

EXAMPLES:: 

sage: A = Zp(7,10) 

sage: S.<x> = A[] 

sage: B.<t> = A.ext(x^2+7) 

sage: B.inertia_subring() 

7-adic Ring with capped relative precision 10 

""" 

return self.ground_ring() 

def residue_class_field(self): 

""" 

Returns the residue class field. 

INPUT: 

- self -- a p-adic ring 

OUTPUT: 

- the residue field 

EXAMPLES:: 

sage: A = Zp(7,10) 

sage: S.<x> = A[] 

sage: B.<t> = A.ext(x^2+7) 

sage: B.residue_class_field() 

Finite Field of size 7 

""" 

return self.ground_ring().residue_class_field() 

def residue_ring(self, n): 

""" 

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

EXAMPLES:: 

sage: S.<x> = ZZ[] 

sage: W.<w> = Zp(5).extension(x^2 - 5) 

sage: W.residue_ring(1) 

Ring of integers modulo 5 

The following requires implementing more general Artinian rings:: 

sage: W.residue_ring(2) 

Traceback (most recent call last): 

... 

NotImplementedError 

""" 

if n == 1: 

return self.ground_ring().residue_ring(1) 

else: 

raise NotImplementedError 

#def discriminant(self, K=None): 

# 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. 

# ## 

# raise NotImplementedError 

#def is_abelian(self): 

# raise NotImplementedError 

#def is_normal(self): 

# raise NotImplementedError 

def gen(self, n=0): 

""" 

Returns a generator for self as an extension of its ground ring. 

EXAMPLES:: 

sage: A = Zp(7,10) 

sage: S.<x> = A[] 

sage: B.<t> = A.ext(x^2+7) 

sage: B.gen() 

t + O(t^21) 

""" 

if n != 0: 

raise IndexError("only one generator") 

return self([0,1]) 

def uniformizer_pow(self, n): 

""" 

Returns the nth power of the uniformizer of self (as an 

element of self). 

EXAMPLES:: 

sage: A = Zp(7,10) 

sage: S.<x> = A[] 

sage: B.<t> = A.ext(x^2+7) 

sage: B.uniformizer_pow(5) 

t^5 + O(t^25) 

""" 

if n is infinity: 

return self(0) 

else: 

return self(1) << n 

def uniformizer(self): 

""" 

Returns the uniformizer of self, ie a generator for the unique 

maximal ideal. 

EXAMPLES:: 

sage: A = Zp(7,10) 

sage: S.<x> = A[] 

sage: B.<t> = A.ext(x^2+7) 

sage: B.uniformizer() 

t + O(t^21) 

""" 

return self.gen() 

def _uniformizer_print(self): 

""" 

Returns a string representation of how the uniformizer of self 

prints. Mainly for internal use. 

EXAMPLES:: 

sage: A = Zp(7,10) 

sage: S.<x> = A[] 

sage: B.<t> = A.ext(x^2+7) 

sage: B._uniformizer_print() 

't' 

""" 

return self.variable_name() 

# def has_pth_root(self): 

# raise NotImplementedError 

# def has_root_of_unity(self, n): 

# raise NotImplementedError