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 resid
ue symbol and R{e}deis 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}deis dihedral extension over $mathbb{Q}$. We also give a cohomological interpretation of our symbols by triple Massey products in Galois cohomology.
This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of view.
