Integral Hopf-Galois structures on degree \(p^2\) extensions of \(p\)-adic fields
This paper gives the definitive classification of Hopf Galois structures on valuation rings of Galois extensions of \(p\)-adic fields of degree \(p^2\), \(p\) prime. Let \(L/K\) be a finite, totally ramified Galois extension of \(p\)-adic fields of degree \(p^2\) with Galois group \(G\), and assume \(K\) contains a primitive \(p\)th root of unity. Let \(e\) be the absolute ramification index of \(K\). Then there are \(p\), respectively \(p^2\) Hopf Galois structures on \(L/K\) if \(G\) is cyclic, respectively elementary abelian. Their structure depends at most on a field extension \(M\) of degree \(p\) over \(K\). The author explicitly describes the corresponding \(K\)-Hopf algebras and their Hopf orders over \(\mathcal O_K\), the valuation ring of \(K\), extending work of \textit{C. Greither} [Math. Z. 210, 37-68 (1992; Zbl 0737.11038)] and the reviewer [New York J. Math. 2, 86-102 (1996; Zbl 0884.11046)]. Using this he determines which of the Hopf orders are realizable, that is, can be the associated order of the valuation ring of a Galois extension of \(K\). The results depend on \(G\), two ``valuation parameters'', \(i\) and \(j\), and a ``unit parameter'' \(v\) (in \(M\)), as well as whether or not \(p = 2\), and break up into seven cases corresponding to various subregions of the square \[ \{ (i,j) \mid 0 \leq i, j, \leq e/p \} . \] This is applied to determine which of the Hopf Galois structures on \(L\) give rise to Hopf Galois structures on \(\mathcal O_L\). The number of such Hopf Galois structures is either 1 or \(p\). The paper concludes by fitting into this landscape two classes of examples: Galois field extensions \(L/K\) of degree \(p^2\) that are classical Kummer extensions, and Kummer extensions for Lubin-Tate formal groups. The latter examples extend work of \textit{M. J. Taylor} [J. Reine Angew. Math. 358, 97-103 (1985; Zbl 0582.12008)] and the author [Galois module structure and Kummer theory for Lubin-Tate formal groups, in Algebraic Number Theory and Diophantine Analysis, ed. by F. Halter-Koch and R. F. Tichy, de Gruyter, Berlin 55-67 (2000; Zbl 0958.11076)].
scientific article; zbMATH DE number 1763595
scientific article