Elementary Abelian Hopf Galois structures and polynomial formal groups.
Suppose that \(L\) is a Galois field extension of a field \(K\) with finite Galois group \(G\) and that \(H\) is a cocommutative Hopf algebra. \(L\) is an \(H\)-Hopf Galois extension if it is a left \(H\) module algebra with the natural map \(j\colon L\otimes_KH\to\text{End}_K(L)\) an isomorphism. \(H=KG\) gives one example of such a structure and the question of classifying all such structures for a given extension \(L\) is the subject of this paper for \(G\) an elementary Abelian \(p\)-group. The problem can be expressed in purely group theoretic terms as that of classifying regular embeddings of \(G\) into \(\text{Hol}(\Gamma)\subseteq\text{Perm}(\Gamma)\) where \(\Gamma\) is a group of the same cardinality as \(G\), \(\text{Perm}(\Gamma)\) is the group of permutations of \(\Gamma\) and \(\text{Hol}(\Gamma)\) is the normalizer of the left regular representation of \(\Gamma\). For \(G=\mathbb{Z}^n_p\), \(p>n\), \(G\) embeds in \(\text{Hol}(\Gamma)\) only if \(G=\Gamma\) and \(\text{Hol}(G)\) embeds in \(\text{GL}_{n+1}(\mathbb{F}_p)\). Using this the author shows that equivalence classes of regular subgroups correspond to linear isomorphism classes of dimension \(n\) polynomial formal group laws over \(\mathbb{F}_p\). From this he deduces that for \(G=\mathbb{Z}^3_p\) there are \(p^6+p^5-p^2\) equivalence classes of regular embeddings and obtains the general lower bound \((p^n-1)(p^n-p)\cdots(p^n-p^{n-2})\) for the number of equivalence classes if \(G=\mathbb{Z}^n_p\) and \(p>n\).
scientific article; zbMATH DE number 2153365