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 R\'{e}dei's dihedral extension of $\mathbb{Q}$ and R\'{e}dei's triple symbol of rational primes. We give examples of R\'{e}dei type extensions $K$ over real quadratic fields. We also give a cohomological interpretation of our symbols in terms of Massey products.