A class number formula for the p-cyclotomic field

Abstract

Let p be an odd prime number and $$ G = Gal(\user2{\mathbb{Q}}(\zeta _p )/\user2{\mathbb{Q}}) $$ . Let $$ \user1{\mathcal{S}}_G $$ be the classical Stickelberger ideal of the group ring $$ \user2{\mathbb{Z}}[G] $$ . Iwasawa [6] proved that the index $$ [\user2{\mathbb{Z}}[G]^ - :\user1{\mathcal{S}}_G ^ - ] $$ equals the relative class number $$ h_p^ - $$ of $$ \user2{\mathbb{Q}}(\zeta _p ) $$ . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal $$ \user1{\mathcal{S}}_H $$ of $$ \user2{\mathbb{Z}}[H] $$ , and studied some of its properties. In this note, we prove that when $$ p \equiv 3 $$ mod 4 and [G : H] = 2, the index $$ [\user2{\mathbb{Z}}[H]:\user1{\mathcal{S}}_H ] $$ equals the quotient $$ h_p^ - /h(\user2{\mathbb{Q}}(\sqrt { - p} )) $$ .