On Vandiver's conjecture
The main theorem of the paper is the following: Let \(p\) be a prime of the form \(n^4 + 5n^3 + 15n^2 + 25n + 25\) and let \(K\) be the quintic subfield of \(\mathbb{Q} (\zeta_p + \zeta_p^{- 1})\). There exists \(N_0\) such that for \(p > N_0\), \(p\) divides the class number \(h_K \) if and only if \(p\) divides a Bernoulli number \(B_j\) with \(j = (p - 1)/5\), \(2(p - 1)\), \(3(p - 1)/5\), \(4(p - 1)/5\). This family of quintic fields was discovered by \textit{E. Lehmer} [Math. Comput. 50, No. 182, 535-541 (1988; Zbl 0652.12004)]. \textit{R. Schoof} and the reviewer [Math. Comput. 50, No. 182, 543-556 (1988; Zbl 0649.12007)] determined a set of fundamental units of \(K\) and gave examples where \(h_K\) has a prime factor larger than \(p\). The interesting aspect of the present theorem is that the divisibility of a Bernoulli number by \(p\) implies that the class number of the real field \(K\) is divisible by \(p\). Usually such criteria apply only to relative class numbers of imaginary fields. The proof constructs a nontrivial homomorphism from the units of \(K\) to \(\mathbb{Z}/p \mathbb{Z}\) that vanishes on the cyclotomic units when \(p\) divides the appropriate Bernoulli number. This technique is also used to study Vandiver's conjecture under various hypotheses.
scientific article; zbMATH DE number 706666
scientific article