Math 721

Spring 2001

Exam 2

 

  1. Let be a field and let  be an irreducible polynomial.

a)      If  is the splitting field of over  and , show that acts transitively on the roots of

b)      Show that if  is irreducible of degree  then  is a transitive subgroup of

 

  1. Find all transitive subgroups of

 

  1. Let  where  is the real root of the polynomial

a)      Show that is not Galois over

b)      Let  be the splitting field of the polynomial  over  Show that  is Galois over  and compute its Galois group.

c)      Express all roots of the polynomial  in terms of radicals.

 

  1. Let be a finite field.

a)      Show that  for some prime

b)      Show that any element in a finite field can be written as the sum of two squares.