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[source]¶
- 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[source]¶
- Bases: - FrobeniusEndomorphism_finite_field- A class implementing Frobenius endomorphism on prime finite fields (i.e. identity map :-). - fixed_field()[source]¶
- Return the fixed field of - self.- OUTPUT: - a tuple \((K, e)\), where \(K\) is the subfield of the domain consisting of elements fixed by - selfand \(e\) 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: 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] - >>> 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]