ETD

Archivio digitale delle tesi discusse presso l'Università di Pisa

Tesi etd-03222011-104820


Tipo di tesi
Tesi di dottorato di ricerca
Autore
CHECCOLI, SARA
URN
etd-03222011-104820
Titolo
On fields of algebraic numbers with bounded local degrees
Settore scientifico disciplinare
MAT/02
Corso di studi
MATEMATICA
Relatori
tutor Prof. Zannier, Umberto
Parole chiave
  • bounded local degrees
  • infinite Galois extensions
  • bounded exponent
Data inizio appello
06/04/2011
Consultabilità
Completa
Riassunto
We provide a characterization of infinite algebraic Galois extensions of the rationals with uniformly bounded local degrees. In particular
we show that that for an infinite Galois extension of the rationals the following three properties are equivalent: having uniformly bounded local degrees at every prime; having uniformly bounded local degrees at almost every prime; having Galois group of finite exponent. The proof of this result enlightens interesting connections with Zelmanov's work on the Restricted Burnside Problem. We give a formula to explicitly compute bounds for the local degrees of an infinite extension in some special cases.
We relate the uniform boundedness of the local degrees to other properties: being a subfield of Q^(d), which is defined as the compositum of all number fields of degree at most d over the rationals; being generated by elements of bounded degree. We prove that the above properties are equivalent for abelian extensions, but not in general; we provide counterexamples based on group-theoretical constructions with extraspecial groups and their modules, for which we give explicit realizations.
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