### 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 language | English (US) |
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Pages (from-to) | 78-83 |

Number of pages | 6 |

Journal | Mathematische Nachrichten |

Volume | 260 |

DOIs | |

State | Published - Jan 1 2003 |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Popescu, A., Popescu, N., & Zaharescu, A. (2003). On the spectral norm of algebraic numbers.

*Mathematische Nachrichten*,*260*, 78-83. https://doi.org/10.1002/mana.200310106