Trace series on \(\tilde {\mathbb Q}_k\)
type: Item
instance of: scholarly article
title: Trace series on \(\tilde {\mathbb Q}_k\)
zbMATH Open document ID: 1060.12004
DOI: 10.1007/BF03322746
author: Angel Popescu
author: N. Popescu
author: Alexandru Zaharescu
published in: Results in Mathematics
publication date: 2003-10-29T00:00:00Z
review text: In [Manuscr. Math. 110, No. 4, 527--541 (2003; Zbl 1027.12004)], the authors generalize the notion of minimal polynomial to a certain class of transcendental elements of \(\widetilde{\overline{\mathbb Q}}\), where \(\widetilde{\overline{\mathbb Q}}\) denotes the completion of the algebraic closure \({\overline{\mathbb Q}}\) of \({\mathbb Q}\) in \({\mathbb C}\) relative to the spectral norm \(\| x\| = \max\{| \sigma x| \mid \sigma \in G\}\) on \({\overline{\mathbb Q}}\), where \(G= \text{ Gal}( {\overline{\mathbb Q}}/{\mathbb Q})\). In the paper under review, for any element of \(\widetilde{\mathbb Q} _ K\), the spectral completion of \(\overline{\mathbb Q}\), now relative to the norm \(\| x\| _ K= \sup\{ | \sigma x| \mid \sigma \in \text{ Gal}(\overline{\mathbb Q}/K)\}\) where \(K\) is a fixed number field containing \(i=\sqrt{-1}\), it is associated a trace series and a trace function. First, the authors introduce \(\widetilde{\mathbb Q} _ K\) as a \({\mathbb C}\)-Banach algebra and prove that it is isomorphic to the \({\mathbb C}\)-Banach algebra of all continuous functions defined on the absolute Galois group \(G _ K =\text{ Gal} (\overline{\mathbb Q}/K)\). Using the Haar measure \(d\sigma\) on the compact group \(G_ K\), they define the trace of an element \(x\) of \(\widetilde{\mathbb Q} _ K\) as \(\text{ Tr}(x):= \int _ {G_ K} \sigma(x) d\sigma\). Next it is defined the trace series of an element \(x\) of \(\widetilde{\mathbb Q} _ K\) as \(\sum _ {j=0} ^ \infty \text{ Tr}(x^ j) z^ j \in {\mathbb C}z\) and also the trace function of \(x\). The main tool in computing the trace series is a measure preserving function \(H\colon G_ K\to [0,1]\). Next, the special case of the function \(f(t) =\exp (-2\pi i mt)+ \exp(2\pi i nt)\) is considered. In the last part, the paper deals with Fourier series in \(\widetilde{\mathbb Q} _ K\) and maximal trace series for elements of \(\widetilde{\mathbb Q} _ K\).
Mathematics Subject Classification ID: 12J05
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R99
Mathematics Subject Classification ID: 11S20
zbMATH DE Number: 1997692
zbMATH Keywords: minimal polynomials
zbMATH Keywords: trace series
zbMATH Keywords: Haar measure
zbMATH Keywords: spectral norm
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
cites work: On the closed subfields of \(C_p\)
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cites work: Q5590141
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cites work: Spectral norms on valued fields
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cites work: On the spectral norm of algebraic numbers
description: scientific article; zbMATH DE number 1997692