Finite field morphisms for prime fields

Special implementation for prime finite field of:

• embeddings of such field into general finite fields

• Frobenius endomorphisms (= identity with our assumptions)

See also

sage.rings.finite_rings.hom_finite_field

AUTHOR:

• Xavier Caruso (2012-06-29)

class sage.rings.finite_rings.hom_prime_finite_field.FiniteFieldHomomorphism_prime

Bases: FiniteFieldHomomorphism_generic

A class implementing embeddings of prime finite fields into general finite fields.

class sage.rings.finite_rings.hom_prime_finite_field.FrobeniusEndomorphism_prime

Bases: FrobeniusEndomorphism_finite_field

A class implementing Frobenius endomorphism on prime finite fields (i.e. identity map :-).

fixed_field()

Return the fixed field of self.

OUTPUT:

• a tuple $(K,e)$, where $K$ is the subfield of the domain consisting of elements fixed by self and $e$ is an embedding of $K$ into the domain.

Note

Since here the domain is a prime field, the subfield is the same prime field and the embedding is necessarily the identity map.

EXAMPLES:

Sage

sage: k.<t> = GF(5)
sage: f = k.frobenius_endomorphism(2); f
Identity endomorphism of Finite Field of size 5
sage: kfixed, embed = f.fixed_field()

sage: kfixed == k
True
sage: [ embed(x) == x for x in kfixed ]
[True, True, True, True, True]

Python

>>> from sage.all import *
>>> k = GF(Integer(5), names=('t',)); (t,) = k._first_ngens(1)
>>> f = k.frobenius_endomorphism(Integer(2)); f
Identity endomorphism of Finite Field of size 5
>>> kfixed, embed = f.fixed_field()

>>> kfixed == k
True
>>> [ embed(x) == x for x in kfixed ]
[True, True, True, True, True]

class sage.rings.finite_rings.hom_prime_finite_field.SectionFiniteFieldHomomorphism_prime

Bases: SectionFiniteFieldHomomorphism_generic