Splitting fields of elements in arithmetic groups

We prove that the number of unimodular integral $n\times n$ matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group $S_n$ is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases.

More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field $F$. Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes developed by Jouve, Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice point counting result, and a non-concentration phenomenon for lattice points in algebraic subvarieties of the group variety, both established previously by the authors.

The results answer a question posed by Rivin and by Jouve, Kowalski and Zywina, who have considered Galois groups of random products of elements in algebraic groups.

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Published 20 February 2012