# Number fields¶

class sage.categories.number_fields.NumberFields(s=None)

Bases: sage.categories.category_singleton.Category_singleton

The category of number fields.

EXAMPLES:

We create the category of number fields:

sage: C = NumberFields()
sage: C
Category of number fields


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()

class ElementMethods
class NumberFields.ParentMethods
NumberFields.super_categories()

EXAMPLES:

sage: NumberFields().super_categories()
[Category of fields]


Monoids

Objects