Fields of algebraic numbers with bounded local degrees and their properties (Q2839393)

In this paper, some properties of infinite algebraic extensions of the field \(\mathbb{Q}\) of rational numbers and the relations between them are studied. The first property considered is the uniform boundedness of the local degrees of an infinite algebraic extension \(K\) of \(\mathbb{Q}\), namely the existence of a constant \(b\), depending only on \(K\), such that, for every prime number \(p\) and every place \(v_p\) of \(K\) which extends the \(p\)-adic one, the completion \(K_{v_p}\) of \(K\) with respect to \(v_p\) is a finite extension of \(\mathbb{Q}_p\) of degree bounded by \(b\). The author proves that for a Galois infinite extension \(K|\mathbb{Q}\) to have the property of uniform boundedness of the local degrees at every prime it is sufficient to have it at almost every prime and that this property is equivalent to \(Gal(K|\mathbb{Q})\) having finite exponent. To this end, the author uses Chebotarev's Density Theorem, a theorem of Shafarevich on the number of generators of the Galois group of a \(p\)-extension of \(p\)-adic fields and Zelmanov's result on the Restricted Burnside Problem.NEWLINENEWLINEThe second property considered (denoted by (b)) is the inclusion of the infinite algebraic extension \(K\) of \(\mathbb{Q}\) in the field \(\mathbb{Q}^{(d)}\), which is defined as the compositum, in a fixed algebraic closure of \(\mathbb{Q}\), of all number fields of degree at most \(d\) over \(\mathbb{Q}\). It was proven in [\textit{E. Bombieri} and \textit{U. Zannier}, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 12, No. 1, 5--14 (2001; Zbl 1072.11077)] that \(\mathbb{Q}^{(d)}\) possesses the uniform boundedness property. Hence this second property implies the first. Besides property (b) is implied by the property denoted by (c), namely that every number field contained in \(K\) can be generated by elements of uniformly bounded degree. The author proves that the three properties: uniform boundedness (denoted by (a)), (b) and (c) are equivalent when \(K|\mathbb{Q}\) is abelian and that the implications (a) \(\Rightarrow\) (b), (b) \(\Rightarrow\) (c) do not hold in general, by providing counterexamples based on group-theoretical constructions with extraspecial groups and their modules for which she gives explicit Galois realizations over adequate cyclotomic fields.