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Hopf-Galois theory was first described by S. U. Chase and M. E. Sweedler [Hopf algebras and Galois theory. Springer, Cham (1969; Zbl 0197.01403)], with the hope that it might shed light on the study of inseparable extensions. However, the theory applies for separable extensions as well, in which case there is a group-theoretic classification of Hopf-Galois structures due to C. Greither and B. Pareigis [J. Algebra 106, 239-258 (1987; Zbl 0615.12026)]. In the setting of Galois extensions, which is the main focus of the book under review, there is a one-to-one correspondence between the Hopf-Galois structures on a Galois G-extension and the regular subgroups normalized by left translations in the symmetric group of G. The type of a Hopf-Galois structure is defined to be the isomorphism class of the associated regular subgroup.Classical Galois module theory is about the study of the Galois module structure of rings of integers in number fields. A famous theorem of Leopoldt states that if L/Q is any abelian G-extension, then the ring of integers O L of L is free over the associated order. However, this does not generalize, even at the local level, when one considers relative or non-abelian extensions. N. P. Byott [in: Algebraic number theory and Diophantine analysis. Proceedings of the international conference, Graz, Austria, 1998. Berlin: Walter de Gruyter. 55-67 (2000; Zbl 0958.11076)] obtained examples of local Galois G-extensions L/K for which O L is not free over the classical associated orderbut is free over the associated order A H of L in some other Hopf-Galois structure H. This interesting result suggests that the consideration of various Hopf-Galois structures, other than the classical Hopf-Galois structure K[G], can broaden the scope of Galois module theory.

Let K be a local field whose residue field has characteristic p and let L/K be a finite separable totally ramified extension of degree n = up ν . Let σ 1 , . . . , σ n denote the K-embeddings of L into a separable closure K sep of K. For 1 ≤ h ≤ n let e h (X 1 , . . . , X n ) denote the hth elementary symmetric polynomial in n variables, and for α ∈ L set E h (α) = e h (σ 1 (α), . . . , σ n (α)). Set j = min{v p (h), ν}. We show that for r ∈ Z we have E h (M r L ) ⊂ M ⌈(i j +hr)/n⌉ K , where i j is the jth index of inseparability of L/K. In certain cases we also show that E h (M r L ) is not contained in any higher power of M K .