This book studies Hopf algebras over valuation rings of local fields and their application to the theory of wildly ramified extensions of local fields. The results, not previously published in book form, show that Hopf algebras play a natural role in local Galois module theory. Included in this work are expositions of short exact sequences of Hopf algebras; Hopf Galois structures on separable field extensions; a generalization of Noether's theorem on the Galois module structure of tamely ramified extensions of local fields to wild extensions acted on by Hopf algebras; connections between tameness and being Galois for algebras acted on by a Hopf algebra; constructions by Larson and Greither of Hopf orders over valuation rings; ramification criteria of Byott and Greither for the associated order of the valuation ring of an extension of local fields to be Hopf order; the Galois module structure of wildly ramified cyclic extensions of local fields of degree p and p 2 ; and Kummer theory of formal groups. Contents: Introduction; Hopf algebras and Galois extensions; Hopf Galois structures on separable field extensions; Tame extensions and Noether's theorem; Hopf algebras of rank p; Larson orders; Cyclic extensions of degree p; Non-maximal orders; Ramification restrictions; Hopf algebras of rank p 2 ; Cyclic Hopf Galois extensions of degree p 2 ; Formal groups; Principal homogeneous spaces and formal groups; Bibliography; Index.

Let L | K L|K be a Galois extension of fields with finite Galois group G G . Greither and Pareigis showed that there is a bijection between Hopf Galois structures on L | K L|K and regular subgroups of P e r m ( G ) Perm(G) normalized by G G , and Byott translated the problem into that of finding equivalence classes of embeddings of G G in the holomorph of groups N N of the same cardinality as G G . In 2007 we showed, using Byott’s translation, that fixed point free endomorphisms of G G yield Hopf Galois structures on L | K L|K . Here we show how abelian fixed point free endomorphisms yield Hopf Galois structures directly, using the Greither-Pareigis approach and, in some cases, also via the Byott translation. The Hopf Galois structures that arise are “twistings” of the Hopf Galois structure by H λ H_{\lambda } , the K K -Hopf algebra that arises from the left regular representation of G G in P e r m ( G ) Perm(G) . The paper concludes with various old and new examples of abelian fixed point free endomorphisms.

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Let L/K be a Galois extension of fields with Galois group Γ, and suppose L/K is also an H-Hopf Galois extension. Using the recently uncovered connection between Hopf Galois structures and skew left braces, we introduce a method to quantify the failure of surjectivity of the Galois correspondence from subHopf algebras of H to intermediate subfields of L/K, given by the Fundamental Theorem of Hopf Galois Theory. Suppose L ⊗ K H = LN where N ∼ = (G, ⋆). Then there exists a skew left brace (G, ⋆, •) where (G, •) ∼ = Γ. We show that there is a bijective correspondence between the set of intermediate fields E between K and L that correspond to K-subHopf algebras of H and a set of subskew left braces of G that we call the •-stable subgroups of (G, ⋆). Counting these subgroups and comparing that number with the number of subgroups of Γ ∼ = (G, •) describes how far the Galois correspondence for the H-Hopf Galois structure is from being surjective. The method is illustrated by a variety of examples.

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