On Triple Quadratic Residue Symbols in Real Quadratic Fields

We introduce triple quadratic residue symbols $[{\mathfrak{p}}_1,{\mathfrak{p}}_2,{\mathfrak{p}}_3]$ for certain finite primes ${\mathfrak{p}}_i$’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 ${\mathfrak{p}}_1,{\mathfrak{p}}_2,{\mathfrak{p}}_3$ and infinite primes, from which we derive mod 2 arithmetic triple Milnor invariants ${\mu }_2(123)$ yielding the triple symbol $[{\mathfrak{p}}_1,{\mathfrak{p}}_2,{\mathfrak{p}}_3]=(-1{)}^{{\mu }_2(123)}$. Our symbols $[{\mathfrak{p}}_1,{\mathfrak{p}}_2,{\mathfrak{p}}_3]$ describes the decomposition law of ${\mathfrak{p}}_3$ in a certain dihedral extension $K$ over $k$ of degree 8, determined by ${\mathfrak{p}}_1,{\mathfrak{p}}_2$. The field $K$ and our symbols $[{\mathfrak{p}}_1,{\mathfrak{p}}_2,{\mathfrak{p}}_3]$ are generalizations over real quadratic fields of Rédei’s dihedral extension of $\mathbb{Q}$ 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.