Math 721

Homework 6

1. Let be algebraic. Let be the set of elements of that are separable over Let the set of elements that are purely inseparable over Show the following:

a) is a separable extension of

b) is a purely inseparable extension of

c) is purely inseparable over

d)

e) is separable over if and only if

f) If is normal over then is Galois over is Galois over and

2. Let be a polynomial with roots and let be the splitting field (over the rationals) of If is the reduction of modulo the prime and the roots of are then show that the splitting field of over is and that (as permutation groups on the roots), may be identified with a subgroup of

3. Compute the Galois groups (over Q) of the following polynomials:

a) (where is not a cube).

b)

c)

d) (where is not a perfect 5^th power).