### Computing the Galois group of a p-adic polynomial

Linfoot Number Theory Seminar

27th March 2019, 11:00 am – 12:00 pm

Howard House, 2nd Floor Seminar Room

We present a family of algorithms for computing the Galois group Gal(F/Q_p) of a polynomial F(x) in Q_p[x] over the p-adics. These are based on the "resolvent method" machinery initially used by Stauduhar (1973) for computing Galois groups of degree up to 7 over the rationals. The run-time of the algorithm essentially depends on the degree of certain resolvent polynomials; we present some novel ways to try to keep these small. Our implementation for Magma is freely available online (https://cjdoris.github.io/pAdicGaloisGroup) and has been used to compute the Galois groups of all extensions of Q_2 of degree up to 22 excluding 16, of which degrees 18, 20 and 22 are new.

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