# Polynomial time algorithms for Galois groups

Computational Group Theory

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## Abstract

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

- (i)
There are polynomial time algorithms to determine:

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

- (b)
the prime divisors of the order of a solvable Galois group

- (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 S

_{n}or A_{n}.

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

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## References

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1984