Finite field morphisms using Givaro

Special implementation for givaro finite fields of:

• embeddings between finite fields

• frobenius endomorphisms

SEEALSO:

:mod:`sage.rings.finite_rings.hom_finite_field`

AUTHOR:

• Xavier Caruso (2012-06-29)

class sage.rings.finite_rings.hom_finite_field_givaro.FiniteFieldHomomorphism_givaro

Bases: sage.rings.finite_rings.hom_finite_field.FiniteFieldHomomorphism_generic

class sage.rings.finite_rings.hom_finite_field_givaro.FrobeniusEndomorphism_givaro

Bases: sage.rings.finite_rings.hom_finite_field.FrobeniusEndomorphism_finite_field

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

The name of the variable used for the subfield (if it is not a prime subfield) is suffixed by _fixed.

EXAMPLES:

sage: k.<t> = GF(5^6)
sage: f = k.frobenius_endomorphism(2)
sage: kfixed, embed = f.fixed_field()
sage: kfixed
Finite Field in t_fixed of size 5^2
sage: embed
Ring morphism:
  From: Finite Field in t_fixed of size 5^2
  To:   Finite Field in t of size 5^6
  Defn: t_fixed |--> 4*t^5 + 2*t^4 + 4*t^2 + t

sage: tfixed = kfixed.gen()
sage: embed(tfixed)
4*t^5 + 2*t^4 + 4*t^2 + t

class sage.rings.finite_rings.hom_finite_field_givaro.SectionFiniteFieldHomomorphism_givaro

Bases: sage.rings.finite_rings.hom_finite_field.SectionFiniteFieldHomomorphism_generic