Coverage for local/lib/python2.7/site-packages/sage/categories/discrete_valuation.py : 24%

r""" 

Discrete Valuation Rings (DVR) and Fields (DVF) 

""" 

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

# 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 sage.categories.euclidean_domains import EuclideanDomains 

from sage.categories.fields import Fields 

class DiscreteValuationRings(Category_singleton): 

""" 

The category of discrete valuation rings 

EXAMPLES:: 

sage: GF(7)[['x']] in DiscreteValuationRings() 

True 

sage: TestSuite(DiscreteValuationRings()).run() 

""" 

def super_categories(self): 

""" 

EXAMPLES:: 

sage: DiscreteValuationRings().super_categories() 

[Category of euclidean domains] 

""" 

return [EuclideanDomains()] 

class ParentMethods: 

@abstract_method 

def uniformizer(self): 

""" 

Return a uniformizer of this ring. 

EXAMPLES:: 

sage: Zp(5).uniformizer() 

5 + O(5^21) 

sage: K.<u> = QQ[[]] 

sage: K.uniformizer() 

u 

""" 

@abstract_method 

def residue_field(self): 

""" 

Return the residue field of this ring. 

EXAMPLES:: 

sage: Zp(5).residue_field() 

Finite Field of size 5 

sage: K.<u> = QQ[[]] 

sage: K.residue_field() 

Rational Field 

""" 

class ElementMethods: 

@abstract_method 

def valuation(self): 

""" 

Return the valuation of this element. 

EXAMPLES:: 

sage: x = Zp(5)(50) 

sage: x.valuation() 

2 

""" 

def euclidean_degree(self): 

""" 

Return the Euclidean degree of this element. 

TESTS:: 

sage: R.<q> = GF(5)[[]] 

sage: (q^3).euclidean_degree() 

3 

sage: R(0).euclidean_degree() 

Traceback (most recent call last): 

... 

ValueError: Euclidean degree of the zero element not defined 

""" 

if not self: 

raise ValueError("Euclidean degree of the zero element not defined") 

return self.valuation() 

def quo_rem(self, other): 

""" 

Return the quotient and remainder for Euclidean division 

of ``self`` by ``other``. 

TESTS:: 

sage: R.<q> = GF(5)[[]] 

sage: (q^2 + q).quo_rem(q) 

(1 + q, 0) 

sage: (q + 1).quo_rem(q^2) 

(0, 1 + q) 

sage: q.quo_rem(0) 

Traceback (most recent call last): 

... 

ZeroDivisionError: Euclidean division by the zero element not defined 

""" 

if not other: 

raise ZeroDivisionError("Euclidean division by the zero element not defined") 

P = self.parent() 

if self.valuation() >= other.valuation(): 

return P(self / other), P.zero() 

else: 

return P.zero(), self 

def is_unit(self): 

""" 

Return True if self is invertible. 

EXAMPLES:: 

sage: x = Zp(5)(50) 

sage: x.is_unit() 

False 

sage: x = Zp(7)(50) 

sage: x.is_unit() 

True 

""" 

return self.valuation() == 0 

def gcd(self,other): 

""" 

Return the greatest common divisor of self and other, 

normalized so that it is a power of the distinguished 

uniformizer. 

""" 

from sage.rings.infinity import Infinity 

val = min(self.valuation(), other.valuation()) 

if val is Infinity: 

return self.parent()(0) 

else: 

return self.parent().uniformizer() ** val 

def lcm(self,other): 

""" 

Return the least common multiple of self and other, 

normalized so that it is a power of the distinguished 

uniformizer. 

""" 

from sage.rings.infinity import Infinity 

val = max(self.valuation(), other.valuation()) 

if val is Infinity: 

return self.parent()(0) 

else: 

return self.parent().uniformizer() ** val 

class DiscreteValuationFields(Category_singleton): 

""" 

The category of discrete valuation fields 

EXAMPLES:: 

sage: Qp(7) in DiscreteValuationFields() 

True 

sage: TestSuite(DiscreteValuationFields()).run() 

""" 

def super_categories(self): 

""" 

EXAMPLES:: 

sage: DiscreteValuationFields().super_categories() 

[Category of fields] 

""" 

return [Fields()] 

class ParentMethods: 

@abstract_method 

def uniformizer(self): 

""" 

Return a uniformizer of this ring. 

EXAMPLES:: 

sage: Qp(5).uniformizer() 

5 + O(5^21) 

""" 

@abstract_method 

def residue_field(self): 

""" 

Return the residue field of the ring of integers of  

this discrete valuation field. 

EXAMPLES:: 

sage: Qp(5).residue_field() 

Finite Field of size 5 

sage: K.<u> = LaurentSeriesRing(QQ) 

sage: K.residue_field() 

Rational Field 

""" 

class ElementMethods: 

@abstract_method 

def valuation(self): 

""" 

Return the valuation of this element. 

EXAMPLES:: 

sage: x = Qp(5)(50) 

sage: x.valuation() 

2 

"""