2025/08/31 by Kuramoto, Atsuki · 1 citation
#11R32 #57M05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2509.00667
We introduce triple quadratic residue symbols [\mathfrakp1, \mathfrakp2, \mathfrakp3] for certain finite primes \mathfrakpi's of a real quadratic field k with trivial narrow class group. For this, we determine a presentation of the Galois group of the maximal pro-2 Galois extension over k unramified outside \mathfrakp1, \mathfrakp2, \mathfrakp3 and infinite primes, from which we derive mod 2 arithmetic triple Milnor invariants μ2(123) yielding the triple symbol [\mathfrakp1, \mathfrakp2, \mathfrakp3] = (-1)μ2(123). Our symbols [\mathfrakp1, \mathfrakp2, \mathfrakp3] describes the decomposition law of \mathfrakp3 in a certain dihedral extension K over k of degree 8, determined by \mathfrakp1, \mathfrakp2. The field K and our symbols [\mathfrakp1, \mathfrakp2, \mathfrakp3] are generalizations over real quadratic fields of Rédei's dihedral extension of ℚ and Rédei's triple symbol of rational primes. We give examples of Rédei type extensions K over real quadratic fields. We also give a cohomological interpretation of our symbols in terms of Massey products.