Tenth Algorithmic Number Theory Symposium ANTS-X University of California, San Diego July 9 – 13, 2012 Tenth Algorithmic Number Theory Symposium (ANTS-X) July 9 – 13, 2012

The complex polynomials P(x) with Gal(P(x)-t) \cong M_{23}

Noam D. Elkies

Abstract: We find the polynomials P of degree 23 over C such that the Galois group of P(x)-t is the Mathieu group M_{23}. This was the last case in Muller's list of exceptional groups that can arise as the Galois groups of P(x)-t over C. To prove that our polynomials have Galois group M_{23}, rather than the alternating group A_{23}, we reduce modulo a large prime and use a consequence of Cebotarev's theorem with an effective bound on the discrepancy that is small enough to make the computation feasible.

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