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 \({[}\mathfrak {p}_1, \mathfrak {p}_2, \mathfrak {p}_3]\) describe 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.
