44 research outputs found
Hopf-Galois structures on separable field extensions of degree
No full textIn 2020, Alabdali and Byott described the Hopf-Galois structures arising on
Galois field extensions of squarefree degree. Extending to squarefree
separable, but not necessarily normal, extensions is a natural next step.
One must consider now the interplay between two Galois groups
and , where is the
Galois closure of . In this paper, we give a characterisation and
enumeration of the Hopf-Galois structures arising on separable extensions of
degree where and are distinct odd primes. This work includes the
results of Byott and Martin-Lyons who do likewise for the special case that
.Comment: 30 pages. arXiv admin note: text overlap with arXiv:2102.05759 by
other authors Author comment: this is known and has been cleared with the
relevant author
A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic
No full textThis is the author accepted manuscript. The final version is available from the publisher via the DOI in this recordLet 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 DS/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 ρ of L with vL(ρ)≡−vL(DS/R)−1(modn) generates L as an H-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions
Nilpotent and abelian Hopf-Galois structures on field extensions
No full textThis is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.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
Sensitive HIV-1 DNA Pol Next-Generation Sequencing for the Characterisation of Archived Antiretroviral Drug Resistance
No full textModern HIV-1 treatment effectively suppresses viral amplification in people living with HIV. However, the persistence of HIV-1 DNA as proviruses integrated into the human genome remains the main barrier to achieving a cure. Next-generation sequencing (NGS) offers increased sensitivity for characterising archived drug resistance mutations (DRMs) in HIV-1 DNA for improved treatment options. In this study, we present an ultra-sensitive targeted PCR assay coupled with NGS and a robust pipeline to characterise HIV-1 DNA DRMs from buffy coat samples. Our evaluation supports the use of this assay for Pan-HIV-1 analyses with reliable detection of DRMs across the HIV-1 Pol region. We propose this assay as a new valuable tool for monitoring archived HIV-1 drug resistance in virologically suppressed individuals, especially in clinical trials investigating novel therapeutic approaches
Galois scaffolds and Galois module structure in extensions of characteristic p local fields of degree p2
No full textThis is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.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 p2 of characteristic p local fields. This condition becomes necessary when we restrict to p = 3. For extensions L/K of degree p2 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)]
Hopf-Galois structures of squarefree degree
No full textThis is the author accepted manuscript. The final version is available from Elsevier via the DOI in this recordLet n be a squarefree natural number, and let G, Γ 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 Γ. We give some examples, including
a full treatment of the case where n is the product of three primes
Counting Hopf-Galois structures on cyclic field extensions of squarefree degree
No full textThis is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.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.The first-named author acknowledges support from The Higher Committee for Education Development in Iraq
Sufficient Conditions for Large Galois Scaffolds
No full textThis is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.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 [BCE]. But since our conditions can be difficult to check, we specialize to elementary abelian extensions and extend the main result of [Eld09] 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 OK that lie in K[G] for G an elementary abelian p-group
On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation
No full textThis is the author accepted manuscript. The final version is available from World Scientific Publishing via the DOI in this record
Hopf-Galois structures on non-normal extensions of degree related to Sophie Germain primes
No full textThis is the author accepted manuscript. The final version is available from Elsevier via the DOI in this recordWe 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 = 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 ≥ 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.London Mathematical SocietyUniversity of Exete
