TR03-064 Authors: Vikraman Arvind, Piyush Kurur

Publication: 8th September 2003 19:57

Downloads: 3347

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Given a polynomial f(X) with rational coefficients as input

we study the problem of (a) finding the order of the Galois group of

f(X), and (b) determining the Galois group of f(X) by finding a small

generator set. Assuming the generalized Riemann hypothesis, we prove

the following complexity bounds:

1. The order of the Galois group of an arbitrary polynomial f(X) in

Z[X] can be computed by a polynomial-time oracle machine with a #P

oracle. Hence, the order can be approximated by a randomized

polynomial-time algorithm with access to an NP oracle.

2. For polynomials f with solvable Galois group we show that the order

can be computed exactly by a randomized polynomial-time algorithm with

access to an NP oracle.

3. For all polynomials f with abelian Galois group we show that a

generator set for the Galois group (as a permutation group acting on

the roots of f) can be computed in randomized polynomial time.

These results also hold for polynomials f\in K[X], where the field K

is specified by giving the minimal polynomial of a primitive element

of K.