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On the Chebotarev invariant of a finite group

Presented by: 
Gareth Tracey
Date: 
Tuesday 14th January 2020 - 11:00 to 12:00
Venue: 
INI Seminar Room 2
Abstract: 
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 dene the Chebotarev invariant precisely, and describe some new results
concerning its asymptotic behaviour.
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons