Coverage for local/lib/python2.7/site-packages/sage/categories/number_fields.py : 41%

r""" 

Number fields 

""" 

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

# Copyright (C) 2005 David Kohel <kohel@maths.usyd.edu> 

# William Stein <wstein@math.ucsd.edu> 

# 2008 Teresa Gomez-Diaz (CNRS) <Teresa.Gomez-Diaz@univ-mlv.fr> 

# 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net> 

# 

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

# http://www.gnu.org/licenses/ 

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

from sage.categories.category_singleton import Category_singleton 

from sage.categories.basic import Fields 

class NumberFields(Category_singleton): 

r""" 

The category of number fields. 

EXAMPLES: 

We create the category of number fields:: 

sage: C = NumberFields() 

sage: C 

Category of number fields 

By definition, it is infinite:: 

sage: NumberFields().Infinite() is NumberFields() 

True 

Notice that the rational numbers `\QQ` *are* considered as 

an object in this category:: 

sage: RationalField() in C 

True 

However, we can define a degree 1 extension of `\QQ`, which is of 

course also in this category:: 

sage: x = PolynomialRing(RationalField(), 'x').gen() 

sage: K = NumberField(x - 1, 'a'); K 

Number Field in a with defining polynomial x - 1 

sage: K in C 

True 

Number fields all lie in this category, regardless of the name 

of the variable:: 

sage: K = NumberField(x^2 + 1, 'a') 

sage: K in C 

True 

TESTS:: 

sage: TestSuite(NumberFields()).run() 

""" 

def super_categories(self): 

""" 

EXAMPLES:: 

sage: NumberFields().super_categories() 

[Category of infinite fields] 

""" 

return [Fields().Infinite()] 

def __contains__(self, x): 

r""" 

Returns True if ``x`` is a number field. 

EXAMPLES:: 

sage: NumberField(x^2+1,'a') in NumberFields() 

True 

sage: QuadraticField(-97,'theta') in NumberFields() 

True 

sage: CyclotomicField(97) in NumberFields() 

True 

Note that the rational numbers QQ are a number field:: 

sage: QQ in NumberFields() 

True 

sage: ZZ in NumberFields() 

False 

""" 

import sage.rings.number_field.number_field_base 

return sage.rings.number_field.number_field_base.is_NumberField(x) 

def _call_(self, x): 

r""" 

Constructs an object in this category from the data in ``x``, 

or throws a TypeError. 

EXAMPLES:: 

sage: C = NumberFields() 

sage: C(QQ) 

Rational Field 

sage: C(NumberField(x^2+1,'a')) 

Number Field in a with defining polynomial x^2 + 1 

sage: C(UnitGroup(NumberField(x^2+1,'a'))) # indirect doctest 

Number Field in a with defining polynomial x^2 + 1 

sage: C(ZZ) 

Traceback (most recent call last): 

... 

TypeError: unable to canonically associate a number field to Integer Ring 

""" 

try: 

return x.number_field() 

except AttributeError: 

raise TypeError("unable to canonically associate a number field to %s"%x) 

class ParentMethods: 

pass 

class ElementMethods: 

pass