4 Cyclotomic units and Kummer logarithms

This chapter treats the plus side of the criterion. The real cyclotomic units generate a finite-index subgroup \(C^+\) of the unit group of \(\mathcal O_{K^+}\), and two facts about it are proved. First, if no Bernoulli numerator in Kummer’s range is divisible by \(p\), then \(C^+\) is \(p\)-saturated in the full unit group, hence the index \([(\mathcal O_{K^+})^\times : C^+]\) is prime to \(p\); the tool here is the Kummer logarithm matrix, whose determinant factors as a diagonal of Bernoulli numbers times a nonvanishing Vandermonde determinant. Second, Sinnott’s prime-conductor index theorem [ 5 ] identifies the \(p\)-divisibility of this index with the \(p\)-divisibility of \(h^+\).