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 Z_p-Kummer extensions L of the maximal p-cyclotomic extension K of a number field 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.