The commutativity of the Galois groups of the maximal unramified pro-p-extensions over the cyclotomic Zp-extensions

AbstractLet p be an odd prime number. For the cyclotomic Zp-extension F∞ of a finite algebraic number field F, we denote by L˜(F∞) the maximal unramified pro-p-extension of F∞. In this paper, using Iwasawa theory and the theory of central class fields, we give necessary conditions for Gal(L˜(F∞)/F∞) to be abelian, and give sufficient conditions in certain special cases