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

摘要

We 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.