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

 

 

  1. 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

 

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

 

a)  (where  is not a cube).

b)

c)

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