Hall 123

On the explicit Galois group of multi quadratic extension over Q.

Karthick Babu

IISER Berhampur

Let $S= \{ a_{1}, a_{2}, \dots, a_{n} \}$ be a finite set of nonzero integers. An arbitrary function $\theta : S \rightarrow \{-1, 1\}$ is called choice of signs for $S$. For an odd prime $p$, we will say that $S$ has \textbf{residue pattern $\theta$ modulo $p$} if
\begin{equation*}
\left(\frac{s}{p}\right) = \theta(s) \quad \forall s \in S,
\end{equation*}
where $\left(\frac{\cdot {p}\right)$ is the Legendre symbol mod $p$. In this talk, we calculate the relative density of the set of primes for which $S$ has residue pattern $\theta$ modulo $p$. As an application, we also calculate the explicit structure of the Galois group
$\Gal{\mathbb{K}/\mathbb{Q}}$,
where $\mathbb{K}= \mathbb{Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots, \sqrt{a_{n}})$.

Note: This is an in-person seminar. Please follow all covid protocols.