On the explicit Galois group of $\mathbb{Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots, \sqrt{a_{n}}, \zeta_{d})$ over $\mathbb{Q}$

TL;DR

This paper determines the explicit Galois group structure of fields obtained by adjoining multiple square roots and a primitive root of unity to the rationals, extending previous work on multi-quadratic fields.

Contribution

It provides a detailed description of the Galois group of fields generated by square roots and roots of unity, generalizing prior results to include cyclotomic extensions.

Findings

Explicit Galois group structure over $Q$ for the field with roots of unity and square roots.

Action of the Galois group on roots of unity and square roots.

Extension of previous multi-quadratic Galois group calculations.

Abstract

Let $S= \{ a_{1}, a_{2}, \dots, a_{n} \}$ be a finite set of non-zero integers. In \cite{KBAM21}, Karthick Babu and Anirban Mukhopadhyay calculated the explicit structure of the Galois group of multi-quadratic field $\mathbb{Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots, \sqrt{a_{n}})$ over $\mathbb{Q}$. For a positive integer $d \geq 3$, $\zeta_{d}$ denotes the primitive $d$-th root of unity. In this paper, we calculate the explicit structure of the Galois group of $\mathbb{Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots, \sqrt{a_{n}}, \zeta_{d})$ over $\mathbb{Q}$ in terms of its action on $\zeta_{d}$ and $\sqrt{a_{i}}$ for $1 \leq i \leq n$.