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We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions $L/K$ of squarefree degree $n$. If $E/K$ is the normal closure of $L/K$ then $G=\mathrm{Gal}(E/K)$ can be viewed as a permutation group of degree $n$. We show that $G$ has derived length at most $4$, but that many permutation groups of squarefree degree and of derived length $2$ cannot occur. We then investigate in detail the case where $n=pq$ where $q \geq 3$ and $p=2q+1$ are both prime. (Thus $q$ is a Sophie Germain prime and $p$ is a safeprime). We list the permutation groups $G$ which can arise, and we enumerate the Hopf-Galois structures for each $G$. There are six such $G$ for which the corresponding field extensions $L/K$ admit Hopf-Galois structures of both possible types., Comment: 26 pages. Proposition 4.12 and Table 5 corrected. Theorem 3.5 removed. Typos fixed and some improvements made to wording. Accepted for Journal of Pure and Applied Algebra

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

We investigate Hopf–Galois structures on a cyclic field extension L / K of squarefree degree n. By a result of Greither and Pareigis, each such Hopf–Galois structure corresponds to a group of order n, whose isomorphism class we call the type of the Hopf–Galois structure. We show that every group of order n can occur, and we determine the number of Hopf–Galois structures of each type. We then express the total number of Hopf–Galois structures on L / K as a sum over factorisations of n into three parts. As examples, we give closed expressions for the number of Hopf–Galois structures on a cyclic extension whose degree is a product of three distinct primes. (There are several cases, depending on congruence conditions between the primes.) We also consider one case where the degree is a product of four primes.

Let $n$ be a squarefree natural number, and let $G$, $\Gamma$ be two groups of order $n$. We determine the number of Hopf-Galois structures of type $G$ admitted by a Galois extension of fields with Galois group isomorphic to $\Gamma$. We give some examples, including a full treatment of the case where $n$ is the product of three primes., Comment: 29 pages; arXiv identifier of our preprint on skew braces now included

Let $n \geq 1$ be a squarefree integer, and let $M$, $A$ be two groups of order $n$. Using our previous results on the enumeration of Hopf-Galois structures on Galois extensions of fields of squarefree degree, we determine the number of skew braces (up to isomorphism) with multiplicative group $M$ and additive group $A$. As an application, we enumerate skew braces whose order is the product of three distinct primes., Comment: 23 pages; arXiv identifier of our preprint on Hopf-Galois structures of squarefree degree now included

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.

— Given an algebraic number field k and a finite group Γ, we write R(Ok[Γ]) for the subset of the locally free classgroup Cl(Ok[Γ]) consisting of the classes of rings of integers ON in tame Galois extensions N/k with Gal(N/k) ∼= Γ. We determine R(Ok[Γ]), and show it is a subgroup of Cl(Ok[Γ]) by means of a description using a Stickelberger ideal and properties of some cyclic codes, when k contains a root of unity of prime order p and Γ = V oC, where V is an elementary abelian group of order pr and C is a cyclic group of order m > 1 acting faithfully on V and making V into an irreducible Fp[C]-module. This extends and refines results of Byott, Greither and Sodaigui for p = 2 in Crelle, respectively of Bruche and Sodaigui for p > 2 in J. Number Theory, which cover only the case m = pr−1 and determine only the image R(M) of R(Ok[Γ]) under extension of scalars from Ok[Γ] to a maximal orderM⊃ Ok[Γ] in k[Γ]. The main result here thus generalizes the calculation of R(Ok[A4]) for the alternating group A4 of degree 4 (the case p = r = 2) given by Byott and Sodaigui in Compositio. Resume. — Etant donne un corps de nombres k et un groupe fini Γ, on note R(Ok[Γ]) le sous-ensemble du groupe de classes localement libre Cl(Ok[Γ]) forme par les classes d’anneaux d’entiers ON d’extensions galoisiennes moderees N/k avec Gal(N/k) ∼= Γ. Nous determinons R(Ok[Γ]), et montrons que c’est un sous-groupe de Cl(Ok[Γ]), au moyen d’une description utilisant un ideal de Stickelberger et des proprietes de certains codes cycliques, lorsque k contient une racine de l’unite d’ordre premier p et Γ = V o C, ou V est un groupe elementaire abelien d’ordre pr et C est un groupe cyclique d’ordre m > 1 agissant fidelement sur V et rendant V un Fp[C]-module irreductible. Ceci generalise et raffine des resultats de Byott, Greither et Sodaigui pour p = 2 dans Crelle, respectivement de Bruche et Sodaigui pour p > 2 dans J. Number Theory, lesquels couvrent seulement le cas m = pr−1 et determinent seulement l’image R(M) de R(Ok[Γ]) sous l’extension des scalaires de Ok[Γ] a un ordre maximalM⊃ Ok[Γ] dans k[Γ]. Le resultat principal ici generalise donc le calcul de R(Ok[A4]) pour le groupe alterne A4 de degre 4 (le cas p = r = 2) donne par Byott et Sodaigui dans Compositio.

Let L/K be a finite separable field extension, and let E be the normal closure of L/K. Let G=Gal(E/K) and G′=Gal(E/L). We call L/K almost cyclic if G′ has a normal cyclic complement in G. This includes the case that L/K is a cyclic Galois extension or a radical extension. We give a method for counting Hopf–Galois structures on an almost cyclic extension L/K. We then count the Hopf–Galois structures on an almost cyclic extension of degree 2n, n⩾3, and determine how many of them are almost classical. This is analogous to a result of T. Kohl [T. Kohl, Classification of the Hopf–Galois structures on prime power radical extensions, J. Algebra 207 (1998) 525–546] which counts the Hopf–Galois structures on a radical extension of odd prime-power degree. In contrast to the odd prime-power degree case, however, we find that an almost cyclic extension L/K of 2-power degree has Hopf–Galois structures for which the Hopf algebra acting on L is not commutative.

Let k be a number field with ring of integers O k , and let Γ be the dihedral group of order 8. For each tame Galois extension N / k with group isomorphic to Γ , the ring of integers O N of N determines a class in the locally free class group Cl ( O k [ Γ ] ) . We show that the set of classes in Cl ( O k [ Γ ] ) realized in this way is the kernel of the augmentation homomorphism from Cl ( O k [ Γ ] ) to the ideal class group Cl ( O k ) , provided that the ray class group of O k for the modulus 4 O k has odd order. This refines a result of the second-named author (J. Algebra 223 (2000) 367–378) on Galois module structure over a maximal order in k [ Γ ] .

Let k be a number field with ring of integers $\mathfrak{O}_k$ , and let $\Gamma=A_4$ be the tetrahedral group. For each tame Galois extension N / k with group isomorphic to $\Gamma$ , the ring of integers $\mathfrak{O}_{N}$ of N determines a class in the locally free class group $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ . We show that the set of classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ realized in this way is the kernel of the augmentation homomorphism from $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ to the ideal class group $\mathrm{Cl}(\mathfrak{O}_k)$ . This refines a result of Godin and Sodaigui ( J. Number Theory 98 (2003), 320–328) on Galois module structure over a maximal order in $k[\Gamma]$ . To the best of our knowledge, our result gives the first case where the set of realizable classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ has been determined for a nonabelian group $\Gamma$ and an arbitrary number field k .

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.

LetVbe a finite additive subgroup of a fieldKof characteristicp>0. We consider sums of the formSh(V:α)=∑v∈V(v+α)hforh≥0 and α∈K. In particular, we give necessary and sufficient conditions for the vanishing ofSh(V; α), in terms of the digit sum in the base-pexpansion ofh, in the case thatVhas indexpinK. The proof involves the polynomialfV(x)=∏v∈V(x−v). We defineV†=fV(K). In the case thatKis finite, the polynomialfV†can be viewed as a generalised trace TrV:K→Vwhich coincides with the usual trace ifVis a subfield ofK.

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 .

For an integer $m \geq 2$, let $\mathcal{P}_m$ be the partition of the unit interval $I$ into $m$ equal subintervals, and let $\mathcal{F}_m$ be the class of piecewise linear maps on $I$ with constant slope $\pm m$ on each element of $\mathcal{P}_m$. We investigate the effect on mixing properties when $f \in \mathcal{F}_m$ is composed with the interval exchange map given by a permutation $\sigma \in S_N$ interchanging the $N$ subintervals of $\mathcal{P}_N$. This extends the work in a previous paper [N.P. Byott, M. Holland and Y. Zhang, DCDS, {\bf 33}, (2013) 3365--3390], where we considered only the "stretch-and-fold" map $f_{sf}(x)=mx \bmod 1$., Comment: 27 pages 6 figures

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 $L/K$ be a finite, totally ramified $p$-extension of complete local fields with residue fields of characteristic $p > 0$, and let $A$ be a $K$-algebra acting on $L$. We define the concept of an $A$-scaffold on $L$, thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where $L/K$ was Galois and $A=K[G]$ for $G=\mathrm{Gal}(L/K)$. When a suitable $A$-scaffold exists, we show how to answer questions generalizing those of classical integral Galois module theory. We give a necessary and sufficient condition, involving only numerical parameters, for a given fractional ideal to be free over its associated order in $A$. We also show how to determine the number of generators required when it is not free, along with the embedding dimension of the associated order. In the Galois case, the numerical parameters are the ramification breaks associated with $L/K$. We apply these results to biquadratic Galois extensions in characteristic 2, and to totally and weakly ramified Galois $p$-extensions in characteristic $p$. We also apply our results to the non-classical situation where $L/K$ is a finite primitive purely inseparable extension of arbitrary exponent that is acted on, via a higher derivation (but in many different ways), by the divided power $K$-Hopf algebra., Further minor corrections and improvements to exposition. Reference [BE] updated. To appear in Ann. Inst. Fourier, Grenoble

We consider the effect on the mixing properties of a piecewise smooth interval map $f$ when its domain is divided into $N$ equal subintervals and $f$ is composed with a permutation of these. The case of the stretch-and-fold map $f(x)=mx \bmod 1$ for integers $m \geq 2$ is examined in detail. We give a combinatorial description of those permutations $\sigma$ for which $\sigma \circ f$ is still (topologically) mixing, and show that the proportion of such permutations tends to $1$ as $N \to \infty$. We then investigate the mixing rate of $\sigma \circ f$ (as measured by the modulus of the second largest eigenvalue of the transfer operator). In contrast to the situation for continuous time diffusive systems, we show that composition with a permutation cannot improve the mixing rate of $f$, but typically makes it worse. Under some mild assumptions on $m$ and $N$, we obtain a precise value for the worst mixing rate as $\sigma$ ranges through all permutations; this can be made arbitrarily close to $1$ as $N → ∞$ (with $m$ fixed). We illustrate the geometric distribution of the second largest eigenvalues in the complex plane for small $m$ and $N$, and propose a conjecture concerning their location in general. Finally, we give examples of other interval maps $f$ for which composition with permutations produces different behaviour than that obtained from the stretch-and-fold map.

Let $G$ be a finite abelian group. A number field $K$ is called a Hilbert-Speiser field of type $G$ if, for every tame $G$-Galois extension $L/K$, the ring of integers $\mathcal{O}_L$ is free as an $\mathcal{O}_K[G]$-module. If $\mathcal{O}_L$ is free over the associated order $\mathcal{A}_{L/K}$ for every $G$-Galois extension $L/K$, then $K$ is called a Leopoldt field of type $G$. It is well known (and easy to see) that if $K$ is Leopoldt of type $G$, then $K$ is Hilbert–Speiser of type $G$. We show that the converse does not hold in general, but that a modified version does hold for many number fields $K$ (in particular, for $K/\mathbb{Q}$ Galois) when $G=C_{p}$ has prime order. We give examples with $G=C_5$ to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.

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

Let S/R be a finite extension of discrete valuation rings of characteristic p>0, and suppose that the corresponding extension L/K of fields of fractions is separable and is H-Galois for some K-Hopf algebra H. Let D_{S/R} be the different of S/R. We show that if S/R is totally ramified and its degree n is a power of p, then any element $\rho$ of L with $v_L(\rho)$ congruent to $-v_L(D_{S/R})-1$ mod n generates L as an H-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. Elder for Galois extensions., Comment: 11 pages; LaTeX. To appear in Journal de Theorie des Nombres de Bordeaux (special volume, Proceedings of Journees Arithmetiques, St Etienne, July 2009). The example in Section 5 has been expanded following a suggestion of the referee. Several minor clarifications have been made to the text

LetL/K be a totally ramified, finite abelian extension of local fields, let\(\mathfrak{O}_L \) and\(\mathfrak{O}\) be the valuation rings, and letG be the Galois group. We consider the powers\(\mathfrak{P}_L^r \) of the maximal ideal of\(\mathfrak{O}_L \) as modules over the group ring\(\mathfrak{O}G\). We show that, ifG has orderp m (withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then\(\mathfrak{P}_L^r \) and\(\mathfrak{P}_L^{r'} \) are isomorphic iffr≡r′ modp m . We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed.

Let p be a prime number, and let K be a finite extension of the rational p-adic field ℚp. Let L/K be a finite abelian extension with Galois group G, and let L, K denote the valuation rings of L, K respectively. Then L is a free module of rank 1 over the group algebra KG. Defining the associated order of the extension L/K byL can be viewed as a module over the ring , and a fortiori over the group ring KG.

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.

Let K be an algebraic number field with ring of integers O and let G be an elementary abelian group of order l k . Let A be a Hopf order in KG and let B be its dual. An order X in a Galois G -extension of K is a semilocal principal homogeneous space over B if X l is a principal homogeneous space over B l and X is integrally closed away from l . We define a map ψ from the group of such X to the locally free classgroup Cl( A ). Assuming that A admits C ≅ F l k × ⊆ Aut( G ), we describe the image of ψ in terms of a Stickelberger ideal in Z C . This generalizes a result of L. R. McCulloh on the classes in Cl( O G ) realized by tame rings of integers.

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.

Planar piecewise isometries with convex polygonal atoms that are piecewise irrational rotations can naturally generate a packing of phase space given by periodic cells that are discs. We show that such packings cannot contain certain subpackings of Apollonian packings, namely those belonging to a family of Arbelos subpackings. We do this by showing that the unit complex numbers giving the directions of tangency within such an isometric-generated packing lie in a finitely generated subgroup of the circle group, whereas this is not the case for the Arbelos subpackings. In the opposite direction, we show that, given an arbitrary disc packing of a polygonal region, there is a piecewise isometry whose regular cells approximate the given packing to any specified precision.