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Let $L/K$ be a finite Galois, totally ramified $p$-extension of complete local fields with perfect residue fields of characteristic $p>0$. In this paper, we give conditions, valid for any Galois $p$-group $G={Gal}(L/K)$ (abelian or not) and for $K$ of either possible characteristic (0 or $p$), that are sufficient for the existence of a Galois scaffold. The existence of a Galois scaffold makes it possible to address questions of integral Galois module structure, which is done in a separate paper. But since our conditions can be difficult to check, we specialize to elementary abelian extensions and extend the main result of [G.G. Elder, Proc. A.M.S. 137 (2009), 1193-1203] from characteristic $p$ to characteristic 0. This result is then applied, using a result of Bondarko, to the construction of new Hopf orders over the valuation ring $\mathfrak{O}_K$ that lie in $K[G]$ for $G$ an elementary abelian $p$-group., Some minor changes to exposition, and references added/updated. To appear in Journal of Number Theory

This paper justifies an assertion in [Eld09] that Galois scaffolds make the questions of Galois module structure tractable. Let k be a perfect field of characteristic p and let K = k((T )). For the class of characteristic p elementary abelian p-extensions L/K with Galois scaffolds described in [Eld09], we give a necessary and sufficient condition for the valuation ring OL to be free over its associated order AL/K in K[Gal(L/K)]. Interestingly, this condition agrees with the condition found by Y. Miyata, concerning a class of cyclic Kummer extensions in characteristic zero.

A Galois scaffold, in a Galois extension of local fields with perfect residue fields, is an adaptation of the normal basis to the valuation of the extension field, and thus can be applied to answer questions of Galois module structure. Here we give a sufficient condition for a Galois scaffold to exist in fully ramified Galois extensions of degree p 2 of characteristic p local fields. This condition becomes necessary when we restrict to p = 3 . For extensions L / K of degree p 2 that satisfy this condition, we determine the Galois module structure of the ring of integers by finding necessary and sufficient conditions for the ring of integers of L to be free over its associated order in K [ Gal ( L / K ) ] .

Let L / K be a finite Galois extension of fields with group Γ. When Γ is nilpotent, we show that the problem of enumerating all nilpotent Hopf–Galois structures on L / K can be reduced to the corresponding problem for the Sylow subgroups of Γ. We use this to enumerate all nilpotent (resp. abelian) Hopf–Galois structures on a cyclic extension of arbitrary finite degree. When Γ is abelian, we give conditions under which every abelian Hopf–Galois structure on L / K has type Γ. We also give a criterion on n such that every Hopf–Galois structure on a cyclic extension of degree n has cyclic type.

Quels invariants d'une p-extension galoisienne de corps local L/K (de corps residuel de characteristique p et groupe de Galois G) determinent la structure des ideaux de L en tant que modules sur l'anneau de groupe Z p [G], Zp l'anneau des entiers p-adiques? Nous considerons cette question dans le cadre des extensions abeliennes elementaires, bien que nous considerions aussi brievement des extensions cycliques. Pour un groupe abelien elementaire G, nous proposons et etudions un nouveau groupe (dans l'anneau de groupe F q [G] ou F q est le corps residuel) ainsi que ses filtrations de ramification.

Let R be a complete discrete valuation ring of mixed characteristic with perfect residue field, and let H be a finite local commutative R-Hopf algebra. We consider when there exists a finite extension of the field of fractions of R, whose valuation ring is a Galois H-object. If this occurs then H is monogenic. Conversely, if H is also cocommutative and H is monogenic, then there exists a valuation ring which is a Galois H-object. To prove this result, we represent H as the kernel of an isogeny of a special type between formal groups over R. We deduce that if A is a finite abelian R-Hopf algebra, such that both A and its dual are local, then A is the associated order of a valuation ring if and only if the dual of A is monogenic.

We explicitly describe, in terms of indecomposable ℤ2[G]-modules, the Galois module structure of ideals in totally ramified biquadratic extensions of local number fields with only one break in their ramification filtration. This paper completeswork begun in [Elder: Canad. J.Math. (5) 50(1998), 1007–1047].

Let L/K be a totally ramified, normal extension of p-adic fields of degree p2. We investigate the behavior of the valuation ring O L in the various Hopf–Galois structures on L/K. Specifically, we determine when O L is Hopf–Galois with respect to a Hopf order in the corresponding Hopf algebra. When this occurs, O L is necessarily a free module over this Hopf order. We also determine which Hopf orders can arise in this way. For cyclic extensions L/K of degree p2, L. N. Childs has shown, under certain restrictions on the ramification numbers, that if O L is Hopf–Galois with respect to a Hopf order in one of the Hopf–Galois structures on L/K, then the same is true in all p Hopf–Galois structures on L/K. We show that this no longer holds if the ramification conditions are relaxed, or if elementary abelian extensions of degree of p2 are considered. We illustrate our results with a special family of Kummer extensions, and with certain extensions arising from Lubin–Tate formal groups.

LetKbe a finite extension of Qp, and suppose thatK/Qpis ramified and that the residue field ofKhas cardinality at least 3. LetK(2)be the second division field ofKwith respect to a Lubin–Tate formal group, and letΓ=Gal(K(2)/K). We determine the associated order inKΓof the valuation ring O(2)ofK(2), and show that O(2)is not free over this order. The integral Galois module structure of certain intermediate fieldsEofK(2)/Kis also considered. In particular, ifp≠2 andKhas residue field of cardinalityporp2, we show that the valuation ring ofEis free over its associated order if and only ifE/Kis either tamely ramified or ap-extension. We also prove that the valuation ring of any weakly ramified abelian extension ofKis free over its associated order.

LetObe the ring of algebraic integers in a number fieldKand letGbe a finite abelian group. McCulloh has characterised the classes in the locally free classgroup Cl(OG) which are realisable by rings of integers of tame normal extensions ofKwith Galois groupG. We extend this to certain extensions which in general need be neither tame nor normal by replacingOGwith a Hopf order A and introducing the strongly A -tame orders as an analogue of tame rings of integers. We also describe the classes realised by principal homogeneous spaces over the dual of A .

Let G, H be Hopf algebras which are finitely generated projective modules over the ground ring R. We use the ideal of integrals of the dual Hopf algebra H* of H to give a new description of Gal(l)(A) where A ∈ Gal(H) and i: G → H is a Hopf algebra morphism which is both monk and epic in the category of finitely generated projective R-modules. We also give a new description of the group operation on Gal(H) for H cocommutative

Let K be a p-adic field, let G and U be groups of order p, and let D (respectively M) be a Hopf order in the group algebra KG (respectively in the algebra of maps U→K). We use the algebraic machinery of [2] to determine the cleft Hopf algebra extensions of M by D, and investigate which of these cleft extensions are Hopf orders in a group algebra

Ramification invariants are necessary, but not in general sufficient, to determine the Galois module structure of ideals in local number field extensions. This insufficiency is associated with elementary abelian extensions, where one can define a refined ramification filtration—one with more ramification breaks [Nigel P. Byott, G. Griffith Elder, New ramification breaks and additive Galois structure, J. Theor. Nombres Bordeaux 17 (1) (2005) 87–107]. The first refined break number comes from the usual ramification filtration and is therefore necessary. Here we study the second refined break number.

We determine all Hopf–Galois structures on a Galois extension of fields of degree pq , where p , q are primes with p≡1 ( mod q) . There are 2 q −1, respectively 2+ p (2 q −3), Hopf–Galois structures when the extension is cyclic, respectively nonabelian. Explicit generators are given for the groups of permutations corresponding to these Hopf–Galois structures.