Monday, December 6 2021
15:30 - 16:30

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
\left(\frac{s}{p}\right) = \theta(s) \quad \forall s \in S,
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
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.

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