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On the Chebotarev invariant of a finite group

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GRA - Groups, representations and applications: new perspectives

Given a nite group X, a classical approach to proving that X is the Galois groupof a Galois extension K=Q can be described roughly as follows: (1) prove that Gal(K=Q) iscontained in X by using known properties of the extension (for example, the Galois group of anirreducible polynomial f(x) 2 Z[x] of degree n embeds into the symmetric group Sym(n)); (2)try to prove that X = Gal(K=Q) by computing the Frobenius automorphisms modulo successiveprimes, which gives conjugacy classes in Gal(K=Q), and hence in X. If these conjugacy classescan only occur in the case Gal(K=Q) = X, then we are done. The Chebotarev invariant of Xcan roughly be described as the eciency of this algorithm”.In this talk we will de ne the Chebotarev invariant precisely, and describe some new resultsconcerning its asymptotic behaviour.

This talk is part of the Isaac Newton Institute Seminar Series series.

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