The index of irregularity of primes (Q1086283)

For an odd prime p, the index of irregularity of p, say I(p), means the number of Bernoulli numbers among \(B_ 2,B_ 4,...,B_{p-3}\) whose numerators are divisible by p. This expository paper discusses the results known about I(p): L. Carlitz's estimate \(I(p)<p/4\) and its generalization by S. Ullom, H. S. Vandiver's congruence \(h^-_ p\equiv 0\) (mod \(p^{I(p)})\), the index formula \([\bar R^-:\bar J^- ]=p^{I(p)}\) by K. Iwasawa and L. Skula, as well as some further results. Here \(h^-_ p\) denotes the relative class number of the p-th cyclotomic field K and \(\bar R^-\) and \(\bar J^-\) are the reductions mod p of the ''minus-parts'' of the group ring \(R={\mathbb{Z}}_ p[G]\) and its Stickelberger ideal J (G being the Galois group of \(K/{\mathbb{Q}}\) and \({\mathbb{Z}}_ p\) the p-adic integers). The proofs make use of p-adic methods, with the Teichmüller character as a common unifying figure.