Massey products and ideal class groups

Abstract

We consider certain Massey products in the cohomology of a Galois extension of fields with coefficients in p-power roots of unity. We prove formulas for these products both in general and in the special case that the Galois extension in question is the maximal extension of a number field unramified outside a set of primes S including those above p and any archimedean places. We then consider those Zp-Kummer extensions L∞ of the maximal p-cyclotomic extension K∞ of a number field K that are unramified outside S. We show that Massey products describe the structure of a certain “decomposition-free” quotient of a graded piece of the maximal unramified abelian pro-p extension of L∞ in which all primes above those in S split completely, with the grading arising from the augmentation filtration on the group ring of the Galois group of L∞/K∞. We explicitly describe examples of the maximal unramified abelian pro-p extensions of unramified outside p Kummer extensions of the cyclotomic field of all p-power roots of unity, for irregular primes p.