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Hopf-Galois Structures on Galois Extensions of Fields of Squarefree Degree
Get PDFHopf-Galois extensions were introduced by Chase and Sweedler [CS69] in 1969, motivated
by the problem of formulating an analogue of Galois theory for inseparable
extensions. Their approach shed a new light on separable extensions. Later in 1987,
the concept of Hopf-Galois theory was further developed by Greither and Pareigis
[GP87]. So, as a problem in the theory of groups, they explained the problem of finding
all Hopf-Galois structures on a finite separable extension of fields. After that, many
results on Hopf-Galois structures were obtained by N. Byott, T. Crespo, S. Carnahan,
L. Childs, and T. Kohl.
In this thesis, we consider Hopf-Galois structures on Galois extensions of squarefree
degree n. We first determine the number of isomorphism classes of groups G of order
n whose centre and commutator subgroup have given orders, and we describe Aut(G)
for each such G. By investigating regular cyclic subgroups in Hol(G), we enumerate
the Hopf-Galois structures of type G on a cyclic extension of fields L/K of degree
n. We then determine the total number of Hopf-Galois structures on L/K. Finally,
we examine Hopf-Galois structures on a Galois extension L/K with arbitrary Galois
group Gamma of order n, and give a formula for the number of Hopf-Galois structures on
L/K of a given type G.The Higher Committee for Education Development in Ira
Integral Hopf–Galois Structures on Degree p2 Extensions of p-adic Fields
No full textAbstractLet L/K be a totally ramified, normal extension of p-adic fields of degree p2. We investigate the behavior of the valuation ring OL in the various Hopf–Galois structures on L/K. Specifically, we determine when OL is Hopf–Galois with respect to a Hopf order in the corresponding Hopf algebra. When this occurs, OL 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 OL 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
On a family of simple skew braces
Get PDFSeveral constructions have been given for families of simple braces, but few
examples are known of simple skew braces which are not braces. In this paper,
we exhibit the first example of an infinite family of simple skew braces which
are not braces and which do not arise from nonabelian simple groups. More
precisely, we show that, for any primes , such that divides
, there are exactly two simple skew braces (up to
isomorphism) of order .Comment: 289 page
Solubility criteria for Hopf-Galois structures
No full textPublishedArticleThis is an open access article. This paper is available via http://nyjm.albany.edu/j/2015/21-40.html.Let be a finite Galois extension of fields with group . Associated to each Hopf-Galois structure on is a group of the same order as the Galois group . The {\em type} of the Hopf-Galois structure is by definition the isomorphism type of . We investigate the extent to which general properties of either of the groups and constrain those of the other. Specifically, we show that if is nilpotent then is soluble, and that if is abelian then is soluble. In contrast to these results, we give some examples where the groups and have different composition factors. In particular, we show that a soluble extension may admit a Hopf-Galois structure of insoluble type
On Insoluble Transitive Subgroups in the Holomorph of a Finite Soluble Group
Get PDFA question of interest both in Hopf-Galois theory and in the theory of skew
braces is whether the holomorph of a finite soluble group
can contain an insoluble regular subgroup. We investigate the more general
problem of finding an insoluble transitive subgroup in
with soluble point stabilisers. We call such a pair irreducible if we
cannot pass to proper non-trivial quotients , of
, so that becomes a subgroup of
. We classify all irreducible solutions of
this problem, showing in particular that every non-abelian composition factor
of is isomorphic to the simple group of order . Moreover, every
maximal normal subgroup of has index .Comment: Revised in line with referee's comments: improvements made to
exposition and minor errors corrected. To appear in Journal of Algebr
Hopf–Galois structures on almost cyclic field extensions of 2-power degree
No full textAbstractLet 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
Integral Galois Module Structure of Some Lubin–Tate Extensions
No full textAbstractLetKbe 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
Galois module structure for dihedral extensions of degree 8: Realizable classes over the group ring
No full textAbstractLet k be a number field with ring of integers Ok, 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 ON of N determines a class in the locally free class group Cl(Ok[Γ]). We show that the set of classes in Cl(Ok[Γ]) realized in this way is the kernel of the augmentation homomorphism from Cl(Ok[Γ]) to the ideal class group Cl(Ok), provided that the ray class group of Ok for the modulus 4Ok 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]
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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