# On Triple Quadratic Residue Symbols in Real Quadratic Fields

## TL;DR

This paper introduces triple quadratic residue symbols in real quadratic fields, linking Galois group presentations, Milnor invariants, and dihedral extensions, generalizing classical Rédéi symbols from rational primes.

## Contribution

It defines new triple residue symbols in real quadratic fields, relates them to Galois groups, Milnor invariants, and dihedral extensions, extending classical Rédéi symbols.

## Key findings

- Defined triple quadratic residue symbols in real quadratic fields.
- Connected symbols to Galois group presentations and Milnor invariants.
- Provided examples of Rédéi type extensions over real quadratic fields.

## Abstract

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.