ON THE VANISHING OF IWASAWA INVARIANTS OF ABSOLUTELY ABELIAN p-EXTENSIONS

ABSTRACT. Let $p $ be any odd prime. We determine all absolutely abelian p-extension fields such that Iwasawa $\lambda_{p}, $ $\mu_{p} $ and $\nu_{p} $-invariants of the cyclotomic $\mathbb{Z}_{p}$-extension are zero, in terms of congruent conditions, pth power residues, and genus fields. 1