On the spectral norm of algebraic numbers

Angel Popescu, Nicolae Popescu, Alexandra Zaharescu

Research output: Contribution to journalArticle

Abstract

In this paper we continue to study the spectral norms and their completions in the case of the algebraic closure ℚ̄ of ℚ in ℂ. Let ℚ̄̃ be the completion of ℚ̄ relative to the spectral norm. We prove that ℚ̄̃ 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 (ℚ̄/ℚ). We prove that any compact, closed to conjugation subset of ℂ is the pseudo-orbit of a suitable element of ℚ̄̃. We also prove that the topological closure of any algebraic number field in ℚ̄̃ is of the form ℚ[x]̃ with x in ℚ̄̃.

Original languageEnglish (US)
Pages (from-to)78-83
Number of pages6
JournalMathematische Nachrichten
Volume260
DOIs
StatePublished - Jan 1 2003
Externally publishedYes

Keywords

  • Banach algebras of continuous functions
  • Galois groups
  • Number fields
  • Spectral norms

ASJC Scopus subject areas

  • Mathematics(all)

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