A note on the index of irregularity

Abstract

In this paper there is shown a “certain addition” to Iwasawa's class number formula for cyclotomic fields: [ R −( l): F −( l)]= l i(l) , where i( l) means the index of irregularity of an odd prime l (i(l) = card{1≤i≤ (l−3) 2 : l B 2i } , B k is the kth Bernoulli number), R −( l) is a certain subgroup of the additive group of the group ring of a cyclic group G of order l−1 over the ring of residue classes mod l and F ( l) is the Stickelberger ideal mod l of the ring R −( l). Special matrices of order (l−1) 2 over the ring of residue classes mod l are considered and it is proved that for rank h of these matrices the relation h + i(l) = (l−1) 2 is valid.