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)
AUTHOR:
- Xavier Caruso (2012-06-29)
-
class
sage.rings.finite_rings.hom_prime_finite_field.
FiniteFieldHomomorphism_prime
¶ Bases:
sage.rings.finite_rings.hom_finite_field.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:
sage.rings.finite_rings.hom_finite_field.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
, where
is the subfield of the domain consisting of elements fixed by
self
andis an embedding of
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: 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]
- a tuple
-
-
class
sage.rings.finite_rings.hom_prime_finite_field.
SectionFiniteFieldHomomorphism_prime
¶ Bases:
sage.rings.finite_rings.hom_finite_field.SectionFiniteFieldHomomorphism_generic