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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 group
of a Galois extension K=Q can be described roughly as follows: (1) prove that Gal(K=Q) is
contained in X by using known properties of the extension (for example, the Galois group of an
irreducible 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 successive
primes, which gives conjugacy classes in Gal(K=Q), and hence in X. If these conjugacy classes
can only occur in the case Gal(K=Q) = X, then we are done. The Chebotarev invariant of X
can roughly be described as the eciency of this \algorithm”.
In this talk we will de ne the Chebotarev invariant precisely, and describe some new results
concerning its asymptotic behaviour.

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

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