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

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