dc.contributor.author Chetcharungkit, Chinnawat dc.date.accessioned 2018-10-01T07:55:39Z dc.date.issued 2018-07-25 dc.description.abstract 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. en_GB dc.description.sponsorship the Royal Thai government en_GB dc.identifier.uri http://hdl.handle.net/10871/34156 dc.language.iso en en_GB dc.publisher University of Exeter en_GB dc.subject Hopf algebras, Integral Hopf-Galois module structures, local fields en_GB dc.title Scaffolds in Non-classical Hopf-Galois Structures en_GB dc.type Thesis or dissertation en_GB dc.date.available 2018-10-01T07:55:39Z dc.contributor.advisor Byott, Nigel dc.publisher.department Mathematics en_GB dc.type.degreetitle PhD in Mathematics en_GB dc.type.qualificationlevel Doctoral en_GB dc.type.qualificationname PhD en_GB
﻿