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

摘要

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