Lecture Notes in Computer Science Volume 174, 1984, pp 225-236
Date: 10 Jun 2005

Polynomial time algorithms for Galois groups

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Abstract

In this paper we present several polynomial time algorithms for Galois groups. We show:

  1. (i)

    There are polynomial time algorithms to determine:

     
  1. (a)

    If the Galois group of an irreducible polynomial over Q is a p-group.

     
  2. (b)

    the prime divisors of the order of a solvable Galois group

     
  3. (ii)

    Using the classification theorem for finite simple groups, there is a polynomial time algorithm to determine whether an irreducible polynomial over Q has Galois group Sn or An.

     

We consider several techniques for computing Galois groups, including the Chebatorev Density Theorem, and their applicability to polynomial time computations.