Classification of the Hopf Galois Structures on Prime Power Radical Extensions

Kohl, Timothy

doi:10.1006/jabr.1998.7479

Cited by 43 publications

(49 citation statements)

References 6 publications

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“…In contrast to the case of odd primes p, there are now noncyclic groups N to consider. Theorem 6.1 is an immediate consequence of Lemmas 6.2 and 6.3 below, whose proofs are adapted from [11]. Lemma 6.2.…”

Section: Some Group Theorymentioning

confidence: 95%

“…Let N be a group of order 2 n , n 3. If Hol(N ) contains an element of order 2 n then N is either the cyclic group C 2 n , the dihedral group D 2 n or the quaternion group Q 2 n This is the analogue for p = 2 of [11,Theorem 4.4]. In contrast to the case of odd primes p, there are now noncyclic groups N to consider.…”

Section: Some Group Theorymentioning

confidence: 95%

“…This gives an alternative approach to Kohl's results [11]. The crucial fact, in both Kohl's approach and ours, is that if N is a group of order p n whose holomorph contains an element of order p n , then N itself must be cyclic [11,Theorem 4.4].…”

Section: Almost Cyclic Extensions Of Odd Prime-power Degreementioning

confidence: 97%

“…So θ 2 is an element of the form (11), and therefore has order at most 2 n−1 . This shows that the exponent of Hol (N ) divides 2 n .…”

Section: Proofmentioning

confidence: 99%

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Hopf–Galois structures on almost cyclic field extensions of 2-power degree

Byott

2007

Journal of Algebra

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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 2 n , 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.

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Section: Some Group Theorymentioning

confidence: 95%

Section: Some Group Theorymentioning

confidence: 95%

Section: Almost Cyclic Extensions Of Odd Prime-power Degreementioning

confidence: 97%

“…So θ 2 is an element of the form (11), and therefore has order at most 2 n−1 . This shows that the exponent of Hol (N ) divides 2 n .…”

Section: Proofmentioning

confidence: 99%

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Hopf–Galois structures on almost cyclic field extensions of 2-power degree

Byott

2007

Journal of Algebra

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“…There may be other K-Hopf algebras H which also endow L/K with a HopfGalois structure. Greither and Pareigis [GP] reduced the determination of all Hopf-Galois structures on a finite separable field extension L/K to a purely group-theoretic problem, which has been solved in various cases (see [B3,CC,C2,K]). …”

Section: Introductionmentioning

confidence: 99%

Integral Hopf–Galois Structures on Degree p2 Extensions of p-adic Fields

Byott¹

2002

Journal of Algebra

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Counting Hopf Galois Structures on Non-Abelian Galois Field Extensions

Carnahan

Childs

1999

Journal of Algebra

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Let L be a field which is a Galois extension of the field K with Galois group G. Greither and Pareigis [GP87] showed that for many G there exist K-Hopf algebras H other than the group ring KG which make L into an H-Hopf Galois extension of K (or a Galois H *object in the sense of Chase and Sweedler [CS69]). Using Galois descent they translated the problem of determining the Hopf Galois structures on L/K into one which depends only on the Galois group G. By this translation, they showed, for example, that any Galois extension with non-abelian G admits at least one non-classical Hopf Galois structure. Byott [By96] further translated the problem to a more amenable group-theoretic problem, and showed that a Galois extension L/K of fields with group G has a unique Hopf Galois structure, namely that by KG, iff n, the order of G, is a Burnside number, that is, is coprime to φ(n), Euler's phi-function of n. (This implies that G is cyclic of square-free order.) The observation of Greither and Pareigis is the only one in the literature to this point which gives any information on the number of Hopf Galois structures on Galois field extensions for G non-abelian. The purpose of this paper is to make a start at determining the number of Hopf Galois structures on L/K for some non-abelian Galois groups G. Before stating our results, we need to describe Byott's counting formula.

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Classification of the Hopf Galois Structures on Prime Power Radical Extensions

Cited by 43 publications

References 6 publications

Hopf–Galois structures on almost cyclic field extensions of 2-power degree

Hopf–Galois structures on almost cyclic field extensions of 2-power degree

Integral Hopf–Galois Structures on Degree p2 Extensions of p-adic Fields

Counting Hopf Galois Structures on Non-Abelian Galois Field Extensions

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