TITLE: Complexity of computing the Galois group
QUESTION [5 upvotes]: There has been some discussion of computing the Galois group of a polynomial over the integers, but I can't seem to find any results, or even a question of what the complexity of this might be. For example, I assume that this is NOT in NP, and IS NP-hard (at least), but am not aware of any reduction...

REPLY [4 votes]: According to a paper Upper bounds on the complexity of some Galois Theory problems by Arvind and Kurur here, a theorem of Landau gives an exponential upper bound in the size of the polynomial under a certain definition of size.  More precisely, her theorem gives a polynomial bound in terms of the size of the Galois group and the size of the polynomial.  But the size of the group can be exponential in the size of the polynomial.  They show if the Galois group is solvable, then the order can be computed by a randomized polynomial time algorithm with an NP oracle.
This paper gives some complexity bounds on computing the order (they show this is in $P^{\# P}$ see the complexity zoo).  Google seems to say there is a paper proving that nilpotence of the Galois group is decidable in polynomial time.