Tenth Algorithmic Number Theory Symposium ANTSX

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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© 201112 Kiran S. Kedlaya (with thanks to Pierrick Gaudry and Emmanuel ThomÃ©)
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