Arithmetic Milnor invariants and multiple power residue symbols in number fields

We introduce arithmetic Milnor invariants and multiple power residue symbols for primes in number fields, following the analogies between primes and knots. Our symbols generalize the Legendre, power residue symbols and the R\'{e}dei triple symbol, and describe the decomposition law of a prime in certain nilpotent extensions of number fields. As a new example, we deal with triple cubic residue symbols by constructing concretely Heisenberg extensions of degree $27$ over the cubic cyclotomic field with prescribed ramification. We also give a cohomological interpretation of our multiple power residue symbols by Massey products in \'{e}tale cohomology.