Trace series on \(\tilde {\mathbb Q}_k\)
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 \(\widetilde{\mathbb Q} _ K\).
scientific article; zbMATH DE number 1997692