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On the spectral norm of algebraic numbers
Author(s) -
Popescu Angel,
Popescu Nicolae,
Zaharescu Alexandru
Publication year - 2003
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310106
Subject(s) - mathematics , combinatorics , norm (philosophy) , algebraic number , physics , mathematical analysis , philosophy , epistemology
In 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} .