Elementary Abelian Hopf Galois Structures and Polynomial Formal Groups

We find a relationship between regular embeddings of G, an elementary abelian p-group of order p, into unipotent upper triangular matrices with entries in Fp and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices. We classify the latter when n = 3 up to linear isomorphism, and use that classification to determine the number of Hopf Galois structures on a Galois extension L/K of fields with Galois group G elementary abelian of order p. We also obtain a lower bound on the number of Hopf Galois structures on a Galois extension L/K when the Galois group is elementary abelian of order p. 1. Hopf Galois structures Let L be a Galois extension of K, fields, with finite Galois group G. Then L is an H-Hopf Galois extension of K for H = KG, where KG acts on L via the natural action of the Galois group G on L. Greither and Pareigis [GP87] showed that for many Galois groups G, L is also an H-Hopf Galois extension of K for H a cocommutative K-Hopf algebra other than KG. The Hopf algebras H that arise have the property that L ⊗K H ≡ LΓ, the group ring over L of a group Γ of the same cardinality as G. We call Γ the associated group of H. From [By96] we know that for H a cocommutative K-Hopf algebra, H-Hopf Galois structures on L correspond bijectively to equivalence classes of regular embeddings β : G → Hol(Γ) ⊂ Perm(Γ). Here Perm(Γ) is the group of permutations of the set Γ, and Hol(Γ) is the normalizer of the left regular representation of Γ in Perm(Γ). Then Hol(Γ) is isomorphic to the semidirect product ΓoAut(Γ). A subgroup J of Hol(G) is regular if |J | = |G| and the projection map π1 : Hol(G) = G o Aut(G)→ G maps J onto G. A homomorphism β : G→ Hol(G) Date: April 1, 2004. 1