On mod 3 triple Milnor invariants and triple cubic residue symbols in the Eisenstein number field

TL;DR

This paper introduces mod 3 triple Milnor invariants and triple cubic residue symbols in the Eisenstein number field, generalizing classical symbols and describing prime decomposition in specific Galois extensions.

Contribution

It develops new invariants and symbols in algebraic number theory, linking them to Galois cohomology and extending classical residue symbols within the Eisenstein number field.

Findings

Generalizes cubic residue symbols and Rédéi's triple symbol

Describes prime decomposition in mod 3 Heisenberg extensions

Provides cohomological interpretation via triple Massey products

Abstract

We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $\mathbb{Q}(\sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and R\'{e}dei's triple symbol, and describes the decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over $\mathbb{Q}(\sqrt{-3})$ with restricted ramification, which we construct concretely in the form similar to R\'{e}dei's dihedral extension over $\mathbb{Q}$. We also give a cohomological interpretation of our symbols by triple Massey products in Galois cohomology.