On a certain nilpotent extension over $\boldsymbol{Q}$ of degree 64 and the 4-th multiple residue symbol

Abstract

In this paper, we introduce the 4-th multiple residue symbol $[p_1,p_2,p_3,p_4]$ for certain four prime numbers $p_i$'s, which extends the Legendre symbol $\big(\frac{p_1}{p_2}\big)$ and the Rédei triple symbol $[p_1,p_2,p_3]$ in a natural manner. For this we construct concretely a certain nilpotent extension $K$ over $\boldsymbol{Q}$ of degree 64, where ramified prime numbers are $p_1, p_2$ and $p_3$, such that the symbol $[p_1,p_2,p_3,p_4]$ describes the decomposition law of $p_4$ in the extension $K/\boldsymbol{Q}$. We then establish the relation of our symbol $[p_1,p_2,p_3,p_4]$ and the 4-th arithmetic Milnor invariant $\mu_2(1234)$ (an arithmetic analogue of the 4-th order linking number) by showing $[p_1,p_2,p_3,p_4] = (-1)^{\mu_2(1234)}$.