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
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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.
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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
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is_aut
()¶ Check if
self
is an automorphismEXAMPLES:
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
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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 ]
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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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