On a certain nilpotent extension over Q of degree 64 and the 4-th multiple residue symbol

2013/11/06 by Fumiya Amano, Amano, Fumiya · 1 citation
Mathematics · #11A15 #11R32 #57M27 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A15 #msc:11R32 #msc:57M27

paper · pdf · doi:10.48550/arxiv.1311.1289

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

arxiv created 2013/11/11 · arxiv updated 2013/11/12

Abstract

In this paper, we introduce the 4-th multiple residue symbol [p₁, p₂, p₃, p₄] for certain four prime numbers p₁, p₂, p₃, p₄, 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 p₁, p₂ and p₃, such that the symbol [p₁, p₂, p₃, p₄] describes the decomposition law of p₄ 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).

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• The decomposition of primes in nonabelian extensions of Heisenberg type and an analogue of Euler's criterion

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