Coverage for local/lib/python2.7/site-packages/sage/categories/complete_discrete_valuation.py : 38%

r""" 

Complete Discrete Valuation Rings (CDVR) and Fields (CDVF) 

""" 

from __future__ import absolute_import 

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

# Copyright (C) 2013 Xavier Caruso <xavier.caruso@normalesup.org> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

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

from sage.misc.abstract_method import abstract_method 

from sage.categories.category_singleton import Category_singleton 

from .discrete_valuation import DiscreteValuationRings, DiscreteValuationFields 

#from sage.misc.cachefunc import cached_method 

class CompleteDiscreteValuationRings(Category_singleton): 

""" 

The category of complete discrete valuation rings 

EXAMPLES:: 

sage: Zp(7) in CompleteDiscreteValuationRings() 

True 

sage: QQ in CompleteDiscreteValuationRings() 

False 

sage: QQ[['u']] in CompleteDiscreteValuationRings() 

True 

sage: Qp(7) in CompleteDiscreteValuationRings() 

False 

sage: TestSuite(CompleteDiscreteValuationRings()).run() 

""" 

def super_categories(self): 

""" 

EXAMPLES:: 

sage: CompleteDiscreteValuationRings().super_categories() 

[Category of discrete valuation rings] 

""" 

return [DiscreteValuationRings()] 

class ElementMethods: 

@abstract_method 

def valuation(self): 

""" 

Return the valuation of this element. 

EXAMPLES:: 

sage: R = Zp(7) 

sage: x = R(7); x 

7 + O(7^21) 

sage: x.valuation() 

1 

""" 

def denominator(self): 

""" 

Return the denominator of this element normalized 

as a power of the uniformizer 

EXAMPLES:: 

sage: K = Qp(7) 

sage: x = K(1/21) 

sage: x.denominator() 

7 + O(7^21) 

sage: x = K(7) 

sage: x.denominator() 

1 + O(7^20) 

Note that the denominator lives in the ring of integers:: 

sage: x.denominator().parent() 

7-adic Ring with capped relative precision 20 

An error is raised when the input is indistinguishable from 0:: 

sage: x = K(0,5); x 

O(7^5) 

sage: x.denominator() 

Traceback (most recent call last): 

... 

ValueError: Cannot determine the denominator of an element indistinguishable from 0 

""" 

return self.parent()(1) 

class CompleteDiscreteValuationFields(Category_singleton): 

""" 

The category of complete discrete valuation fields 

EXAMPLES:: 

sage: Zp(7) in CompleteDiscreteValuationFields() 

False 

sage: QQ in CompleteDiscreteValuationFields() 

False 

sage: LaurentSeriesRing(QQ,'u') in CompleteDiscreteValuationFields() 

True 

sage: Qp(7) in CompleteDiscreteValuationFields() 

True 

sage: TestSuite(CompleteDiscreteValuationFields()).run() 

""" 

def super_categories(self): 

""" 

EXAMPLES:: 

sage: CompleteDiscreteValuationFields().super_categories() 

[Category of discrete valuation fields] 

""" 

return [DiscreteValuationFields()] 

class ElementMethods: 

@abstract_method 

def valuation(self): 

""" 

Return the valuation of this element. 

EXAMPLES:: 

sage: K = Qp(7) 

sage: x = K(7); x 

7 + O(7^21) 

sage: x.valuation() 

1 

""" 

def denominator(self): 

""" 

Return the denominator of this element normalized 

as a power of the uniformizer 

EXAMPLES:: 

sage: K = Qp(7) 

sage: x = K(1/21) 

sage: x.denominator() 

7 + O(7^21) 

sage: x = K(7) 

sage: x.denominator() 

1 + O(7^20) 

Note that the denominator lives in the ring of integers:: 

sage: x.denominator().parent() 

7-adic Ring with capped relative precision 20 

An error is raised when the input is indistinguishable from 0:: 

sage: x = K(0,5); x 

O(7^5) 

sage: x.denominator() 

Traceback (most recent call last): 

... 

ValueError: Cannot determine the denominator of an element indistinguishable from 0 

""" 

if self == 0: 

raise ValueError("Cannot determine the denominator of an element indistinguishable from 0") 

val = self.valuation() 

R = self.parent().integer_ring() 

if val >= 0: 

return R(1) 

else: 

return R(1) << (-val)