Relative normal bases of positive characteristic
type: Item
instance of: scholarly article
title: Relative normal bases of positive characteristic
zbMATH Open document ID: 1020.11069
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2003-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/228283
review text: This paper deals with integral normal bases (INB) of tame ``cyclotomic'' extensions \(K\) of the rational function field \(Q=\mathbb{F}_q(T)\). More precisely, the extension \(K\) is obtained by adjoining the \(P\)-division points of the Carlitz module to \(Q\), for any irreducible polynomial \(P\in\mathbb{F}_q[T]\), and the issue is whether an intermediate layer \(N/k\), \(Q\subset k \subset N \subset k\), admits an INB, that is, whether the projective \(O_k[\Delta]\)-module \(O_N\) is free. (Here \(\Delta\) is the Galois group of \(N/k\).) The extreme case \(K/Q\) is well understood, just like its counterpart in characteristic zero: INB's do exist and are afforded by Thakur's analogue of Gauss sums. Lowering the top field from \(K\) to \(N\) preserves the existence of an INB, but the effect of raising the bottom field from \(Q\) to \(k\) is surprisingly difficult to understand. In the number field setting \(Q\) is replaced by the rationals \(\mathbb{Q}\) and \(K\) by \(\mathbb{Q}(\zeta_p)\), where \(p\) is an odd prime. In this context, Brinkhuis showed that the only choice of \(k\) such that \([K:k]\) is prime and \(K/k\) has an INB is \(k=K^+\). The author of the present paper proves results for the function field case which point in the same direction. In Theorems 3.5 and 4.6 he gives several sufficient conditions for \(N/k\) to be without INB. As an application, he obtains the following interesting consequences: If the characteristic of \(Q\) is not 2 and \(k\) is the quadratic subfield of \(K/Q\), then \(K/k\) has INB if and only if the polynomial \(P\in\mathbb{F}_q[T]\) defining \(K/Q\) is of degree at most 2. The proof of Theorem 3.5 uses the idea of Brinkhuis that the injectivity of a certain map provides a cohomological obstruction for the existence of integral normal bases, while the proof of 4.6 is more explicit, actually showing the nonprincipality of certain resolvent ideals. In the number field case discussed above, the reviewer, \textit{D. Replogle}…
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 1943324
zbMATH Keywords: integral normal bases
zbMATH Keywords: tame ``cyclotomic'' extensions
zbMATH Keywords: rational function field
author: Bruno Anglès
reviewed by: Cornelius Greither
MaRDI profile type: Publication
OpenAlex ID: W2325787634
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DOI: 10.5802/JTNB.343
description: scientific article; zbMATH DE number 1943324