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Tohoku Mathematical Journal
2014
December
SECOND SERIES VOL. 66, NO. 4
Tohoku Math. J.
66 (2014), 501522

Title
ON A CERTAIN NILPOTENT EXTENSION OVER $\boldsymbol{Q}$ OF DEGREE 64 AND THE 4TH MULTIPLE RESIDUE SYMBOL
Author
Fumiya Amano
(Received May 25, 2012, revised October 30, 2013) 
Abstract.
In this paper, we introduce the 4th 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 4th arithmetic Milnor invariant $\mu_2(1234)$ (an arithmetic analogue of the 4th order linking number) by showing $[p_1,p_2,p_3,p_4] = (1)^{\mu_2(1234)}$.
Mathematics Subject Classification.
Primary 11A15; Secondary 11R32, 57M27.
Key words and phrases.
Rédei triple symbol, Milnor invariant, 4th multiple residue symbol.


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