Statement of the main result Sample Clauses

Statement of the main result. Following Movahhedi and Xxxxxx Xxxxx Xx [32], a number field is said to be ‘p-rational’ if the Galois group of its maximal pro-p extension that is unramified outside p is a free pro-p group. It is known that a number field K has this property if and only if it both validates K Xxxxxxxx’s Conjecture at p and is such that Ap is torsion-free (see Xxxxxxx and Xxxxxx Xxxxx Do [24, Theorem 1.2]). We further recall that any K for which both K(ζp) has a unique p-adic place and class number coprime to p has these properties. In particular, if p is regular, then any abelian extension of Q of p-power conductor is p-rational (see [24, Corollary 1.3(ii)]). As another application of our methods, in this chapter we will prove the following result (which does not assume that G is cyclic). In the following we write Z(G) for the centre of a group G.

Statement of the main result. In order to state the main result of this section we fix a topological generator γQ of ΓQ. Noting that Gn/Hn can be identified with the (unique) quotient of ΓQ of order pn—1 we also fix a generating element γn of Gn that projects to the same element of Gn/Hn as γQ. γQ We write χQ for the cyclotomic character of ΓQ and define an element ϵ Σ := n γQ ψ∈G∗n ϵψ eψ of Qp[Gn]× by setting for each ψ in G∗n ϵ ψ γQ :=