On a certain nilpotent extension over 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 $p_1,p_2,p_3,p_4$, which extends the Legendre symbol and the Rédei triple symbol in a natural manner. For this we construct concretely a certain nilpotent extension K over 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/Q. We then establish the relation of our symbol and the 4-th arithmetic Milnor invariant (an arithmetic analogue of the 4-th order linking number).

26 pages, 4 figures; to appear in Tohoku Math. J., figures inserted properly in v2

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