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# Homset for Finite Fields

This is the set of all field homomorphisms between two finite fields.

EXAMPLES:

    sage: R.<t> = ZZ[]
    sage: E.<a> = GF(25, modulus = t^2 - 2)
    sage: F.<b> = GF(625)
    sage: H = Hom(E, F)
    sage: f = H([4*b^3 + 4*b^2 + 4*b]); f
    Ring morphism:
      From: Finite Field in a of size 5^2
      To:   Finite Field in b of size 5^4
      Defn: a |--> 4*b^3 + 4*b^2 + 4*b
    sage: f(2)
    2
    sage: f(a)
    4*b^3 + 4*b^2 + 4*b
    sage: len(H)
    2
    sage: [phi(2*a)^2 for phi in Hom(E, F)]
    [3, 3]

We can also create endomorphisms:

    sage: End(E)
    Automorphism group of Finite Field in a of size 5^2
    sage: End(GF(7))[0]
    Ring endomorphism of Finite Field of size 7
      Defn: 1 |--> 1
    sage: H = Hom(GF(7), GF(49, 'c'))
    sage: H[0](2)
    2

class `sage.rings.finite_rings.homset.``FiniteFieldHomset`(R, S, category=None)

  Bases: `sage.rings.homset.RingHomset_generic`

  Set of homomorphisms with domain a given finite field.

  `index`(item)

    Return the index of `self`.

    EXAMPLES:

        sage: K.<z> = GF(1024)
        sage: g = End(K)[3]
        sage: End(K).index(g) == 3
        True

  `is_aut`()

    Check if `self` is an automorphism

    EXAMPLES:

        sage: Hom(GF(4, 'a'), GF(16, 'b')).is_aut()
        False
        sage: Hom(GF(4, 'a'), GF(4, 'c')).is_aut()
        False
        sage: Hom(GF(4, 'a'), GF(4, 'a')).is_aut()
        True

  `list`()

    Return a list of all the elements in this set of field homomorphisms.

    EXAMPLES:

        sage: K.<a> = GF(25)
        sage: End(K)
        Automorphism group of Finite Field in a of size 5^2
        sage: list(End(K))
        [Ring endomorphism of Finite Field in a of size 5^2
          Defn: a |--> 4*a + 1,
         Ring endomorphism of Finite Field in a of size 5^2
          Defn: a |--> a]
        sage: L.<z> = GF(7^6)
        sage: [g for g in End(L) if (g^3)(z) == z]
        [Ring endomorphism of Finite Field in z of size 7^6
          Defn: z |--> z,
         Ring endomorphism of Finite Field in z of size 7^6
          Defn: z |--> 5*z^4 + 5*z^3 + 4*z^2 + 3*z + 1,
         Ring endomorphism of Finite Field in z of size 7^6
          Defn: z |--> 3*z^5 + 5*z^4 + 5*z^2 + 2*z + 3]

    Between isomorphic fields with different moduli:

        sage: k1 = GF(1009)
        sage: k2 = GF(1009, modulus="primitive")
        sage: Hom(k1, k2).list()
        [
        Ring morphism:
          From: Finite Field of size 1009
          To:   Finite Field of size 1009
          Defn: 1 |--> 1
        ]
        sage: Hom(k2, k1).list()
        [
        Ring morphism:
          From: Finite Field of size 1009
          To:   Finite Field of size 1009
          Defn: 11 |--> 11
        ]

        sage: k1.<a> = GF(1009^2, modulus="first_lexicographic")
        sage: k2.<b> = GF(1009^2, modulus="conway")
        sage: Hom(k1, k2).list()
        [
        Ring morphism:
          From: Finite Field in a of size 1009^2
          To:   Finite Field in b of size 1009^2
          Defn: a |--> 290*b + 864,
        Ring morphism:
          From: Finite Field in a of size 1009^2
          To:   Finite Field in b of size 1009^2
          Defn: a |--> 719*b + 145
        ]

    TESTS:

    Check that trac ticket #11390 is fixed:

        sage: K = GF(1<<16,'a'); L = GF(1<<32,'b')
        sage: K.Hom(L)[0]
        Ring morphism:
          From: Finite Field in a of size 2^16
          To:   Finite Field in b of size 2^32
          Defn: a |--> b^29 + b^27 + b^26 + b^23 + b^21 + b^19 + b^18 + b^16 + b^14 + b^13 + b^11 + b^10 + b^9 + b^8 + b^7 + b^6 + b^5 + b^2 + b

  `order`()

    Return the order of this set of field homomorphisms.

    EXAMPLES:

        sage: K.<a> = GF(125)
        sage: End(K)
        Automorphism group of Finite Field in a of size 5^3
        sage: End(K).order()
        3
        sage: L.<b> = GF(25)
        sage: Hom(L, K).order() == Hom(K, L).order() == 0
        True

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