Linking Invariants for Valuations and Orderings on Fields

Abstract

The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of $\mathbb{Q}$ with unitriangular Galois groups, and contain the Legendre and Redei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the q-th roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient, and relate it to Massey product elements in Galois cohomology.