On the expected number of random primes required to generate a Galois group

Gareth Tracey (Oxford University)

Wednesday 11th November, 2020 16:00-17:00 zoom (online)

Abstract

Given a finite group $X$, a classical approach to proving that $X$ is the Galois group of a Galois extension $K/\mathbb{Q}$ can be described roughly as follows: (1) prove that $\Gal(K/\mathbb{Q})$ is contained in $X$ by using known properties of the extension (for example, the Galois group of an irreducible polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ embeds into the symmetric group $\Sym(n)$); (2) try
to prove that $X = \Gal(K/\mathbb{Q})$ by computing the Frobenius automorphisms modulo successive primes, which gives conjugacy classes in $\Gal(K/\mathbb{Q})$, and hence in $X$. If these conjugacy classes can only occur in the case $\Gal(K/\mathbb{Q})=X$, then we are done.

Although better algorithms exist in practice, this approach has led to some recent breakthroughs in the problem of finding a Galois extension of number fields with a given Galois group. In this talk we will describe some of these new results, and the link to a new and rapidly developing area of finite group theory.

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