Finite fields

class sage.categories.finite_fields.FiniteFields(base_category)

Bases: CategoryWithAxiom_singleton

The category of finite fields.

EXAMPLES:

Sage

sage: K = FiniteFields(); K
Category of finite enumerated fields

Python

>>> from sage.all import *
>>> K = FiniteFields(); K
Category of finite enumerated fields

A finite field is a finite monoid with the structure of a field; it is currently assumed to be enumerated:

Sage

sage: K.super_categories()
[Category of fields,
 Category of finite commutative rings,
 Category of finite enumerated sets]

Python

>>> from sage.all import *
>>> K.super_categories()
[Category of fields,
 Category of finite commutative rings,
 Category of finite enumerated sets]

Some examples of membership testing and coercion:

Sage

sage: FiniteField(17) in K
True
sage: RationalField() in K
False
sage: K(RationalField())
Traceback (most recent call last):
...
TypeError: unable to canonically associate a finite field to Rational Field

Python

>>> from sage.all import *
>>> FiniteField(Integer(17)) in K
True
>>> RationalField() in K
False
>>> K(RationalField())
Traceback (most recent call last):
...
TypeError: unable to canonically associate a finite field to Rational Field

class ElementMethods

Bases: object

class ParentMethods

Bases: object

extra_super_categories()

Any finite field is assumed to be endowed with an enumeration.