  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 R\'{e}dei 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).
