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Title: Scaffolds in non-classical Hopf-Galois structures
Author: Chetcharungkit, Chinnawat
ISNI:       0000 0004 7427 3415
Awarding Body: University of Exeter
Current Institution: University of Exeter
Date of Award: 2018
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For an extension of local fields, a scaffold is shown to be a powerful tool for dealing with the problem of the freeness of fractional ideals over their associated orders (Byott, Childs and Elder: \textit{Scaffolds and Generalized Integral Galois Module Structure}, Ann. Inst. Fourier, 2018). The first class of field extensions admitting scaffolds is \enquote*{near one-dimensional elementary abelian extension}, introduced by Elder (\textit{Galois Scaffolding in One-dimensional Elementary Abelian Extensions}, Proc. Amer. Math. Soc. 2009). However, the scaffolds constructed in Elder's paper arise only from the classical Hopf-Galois structure. Therefore, the study in this thesis aims to investigate scaffolds in non-classical Hopf-Galois structures. Let $L/K$ be a near one-dimensional elementary abelian extension of degree $p^2$ for a prime $p \geq 3.$ We show that, among the $p^2-1$ non-classical Hopf-Galois structures on the extension, there are only $p-1$ of them for which scaffolds may exist, and these exist only under certain restrictive arithmetic condition on the ramification break numbers for the extension. The existence of scaffolds is beneficial for determining the freeness status of fractional ideals of $\mathfrak{O}_L$ over their associated orders. In almost all other cases, there is no fractional ideal which is free over its associated order. As a result, scaffolds fail to exist.
Supervisor: Byott, Nigel Sponsor: Royal Thai Government
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Hopf algebras ; Integral Hopf-Galois module structures ; local fields