On the spectral norm of algebraic numbers

Abstract

AbstractIn this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure \documentclass{article} \usepackage{amssymb} \pagestyle{empty} \begin{document} $ \overline {\mathbb Q} $ \end{document} of ℚ in ℂ. Let \documentclass{article} \usepackage{amssymb} \pagestyle{empty} \begin{document} $ \widetilde{\overline{\mathbb{Q}}} $ \end{document} be the completion of \documentclass{article} \usepackage{amssymb} \pagestyle{empty} \begin{document} $ \overline {\mathbb Q} $ \end{document} relative to the spectral norm. We prove that \documentclass{article} \usepackage{amssymb} \pagestyle{empty} \begin{document} $ \widetilde{\overline{\mathbb{Q}}} $ \end{document} can be identified with the R‐subalgebra of all symmetric functions of C(G), where C(G) denotes the ℂ‐Banach algebra of all continuous functions defined on the absolute Galois group G = Gal\documentclass{article} \usepackage{amssymb} \pagestyle{empty} \begin{document} $ {\overline {\mathbb Q}} / {\mathbb Q} $ \end{document}. We prove that any compact, closed to conjugation subset of ℂ is the pseudo‐orbit of a suitable element of \documentclass{article} \usepackage{amssymb} \pagestyle{empty} \begin{document} $ \widetilde{\overline{\mathbb{Q}}} $ \end{document}. We also prove that the topological closure of any algebraic number field in \documentclass{article} \usepackage{amssymb} \pagestyle{empty} \begin{document} $ \widetilde{\overline{\mathbb{Q}}} $ \end{document} is of the form \documentclass{article} \usepackage{amssymb} \pagestyle{empty} \begin{document} $\widetilde{\mathbb{Q}[x]}$ \end{document} with x in \documentclass{article} \usepackage{amssymb} \pagestyle{empty} \begin{document} $ \widetilde{\overline{\mathbb{Q}}} $ \end{document}.