Hopf-Galois Structures on Cyclic Field Extensions of Squarefree Degree

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1 Hopf-Galois Structures on Cyclic Field Extensions of Squarefree Degree 1 University of Exeter, UK 23/06/ aaab201@exeter.ac.uk

2 Hopf-Galois structure For normal separable field extension L/K, Γ = Gal(L/K) = group of K-automorphisms of L. The group algebra K[G] = { γ Γ C γγ : C γ K} acts on L with the maps below is a Hopf algebra. comultiplication : counit ɛ : antipode S : Cγ γ C γ γ γ K[G] K[G] Cγ γ C γ Cγ γ C γ γ 1 We can generalise this to look at other Hopf algebras H giving L a Hopf-Galois structure.

3 Greither and Pareigis theory Hopf-Galois theory for separable extensions. The key result of [GP87]: Hopf-Galois structures on L/K correspond to regular subgroups G Perm(Γ) normalised by left translations by Γ.

4 Regular embedding Introduction H Perm(Γ) is said to be regular if it satisfies any two of the following conditions which imply to satisfy the other: 1 Stab H (γ) is the trivial group, for any γ Γ 2 H acts transitively on Γ 3 H = Γ. Let β : Γ Hol(G) where Hol(G) = G Aut(G) Perm(G), then β is said to be regular embedding if: 1 β is a group homomorphism 2 β is injective 3 im(β) = β(γ) regular Perm(G). Hol(G) := {[g, α] g G, α Aut(G)}, with the multiplication [g, α][g, α ] = [gα(g ), αα ]

5 Groups of squarefree order For n squarefree any group of order n has form G(d, e, k) = ρ, π : ρ e = 1 = π d : πρπ 1 = ρ k where n = de, gcd(k, e) = 1 and ord e (k) = d. In this case G(d, e, k) isomorphic to G(d, e, k ) if and only if d = d, e = e, and k, k generate the same cyclic subgroup of U(e) the group of units in the ring Z/eZ of integers modulo e. [MM84, Lemma 3.5 & 3.6]. Centre of G(d, e, k) = C z where z = gcd(e, k 1) with e = zg, in which g (respectively, z) is the order of the commutator subgroup G (respectively, the centre Z(G)) of G.

6 Main results Introduction Theorem 1 Let L/K be a cyclic extension of fields of squarefree degree n, and let G(d, e, k) be any group of order n. Let z, g, d as above. Then L/K admits precisely 2 ω(g) φ(d) Hopf-Galois structures of type G, where ω(g) is the number of (distinct) prime factors of g and φ is Euler s totient function. Theorem 2 The total number of Hopf-Galois structures on a cyclic field extension of squarefree degree n is (p v(p, g) 1), dgz=n 2 ω(g) µ(z) p d where the product is over ordered triples (d, g, z) of natural numbers with dgz = n. Here µ is the Möbius function and v(p, g) is the number of distinct prime factors q of g with q 1 (mod p).

7 Counting regular subgroups of type C n in Hol(G) Let G(d, e, k) be a group of squarefree order n. Then Aut(G) is generated by the automorphism θ and automorphism φ s for each s U(e), where θ(ρ) = ρ, θ(π) = ρ z π, φ s (ρ) = ρ s, φ s (π) = π, φ s θφ 1 s = θ s. Therefore, we have Aut(G) = C g U(e) and Aut(G) = gφ(e).

8 Some properties Introduction Proposition 3 Let C be a cyclic subgroup of Hol(G) which is regular on G. Then C is generated by some element x = [ρ a π, θ c φ s ], in which π occurs with exponent 1. In fact, C contains precisely φ(e) generators of this type. We have got x j = [ρ A(j) π j, θ cs(s, j) φ s j ], for j 0. Where S(s, j) = j 1 i=0 si, A(j) = as(sk, j) + czkt (k, s, j), T (k, s, j) = j 1 i=0 S(s, i)ki 1 Proposition 4 Let n = dgz be squarefree. Then the number of isomorphism types of groups G of order n with z and g as above is φ(d) ( g ) 1 µ (p v(p, f ) 1). f f g p d

9 Sketch of proof of Theorem 1 For x to be regular we need: for each q z, s 1 (mod q) and q a; for each q g, either s 1 (mod q) and c 0 (mod q), or s k 1 (mod q) and (s 1)a + cz 0 (mod q). So 2 ω(g) choices for s mod e, φ(z)g possibilities for a mod e, φ(g) possibilities for c mod g. 2 ω(g) φ(e)g choices for x. 2 ω(g) g regular subgroups in Hol(G). Depending on [Byo96, Proposition 1], we have Aut(Cn) Aut(G) 2ω(g) g = 2 ω(g) φ(d) Hopf-Galois structures of type G(d, e, k).

10 Sketch of proof of Theorem 2 For each factorisation n = dgz, we have seen in Proposition 4 that the number of corresponding isomorphism types of group G is φ(d) ( g ) 1 µ (p v(p, f ) 1). f f g p d We have also seen in Theorem 1 that the number of Hopf-Galois structures of each of these types is 2 ω(g) φ(d). To obtain the total number of Hopf-Galois structures, we simply multiply these two quantities and sum over factorisations of n, which after some simplifications give the total number (p v(p, g) 1). dgz=n 2 ω(g) µ(z) p d

11 References Introduction Nigel P. Byott. Uniqueness of Hopf Galois structure for separable field extensions. Comm. Algebra, 24(10): , Cornelius Greither and Bodo Pareigis. Hopf-Galois theory for separable field extensions. Journal of Algebra, 106(1): , M. Ram Murty and V. Kumar Murty. On groups of squarefree order. Mathematlsche Annalen, 267: , 1984.

12 Thank You