We know that is irreducible over for every prime . Suppose is a zero of and consider the field .
(a) Show that are distinct zeros of and show that they are all the zeros of .
(b) Show that is abelian of order .
(c) Show that the fixed field of is .
For (a), these are the nth roots of unity? For (b), consider two automorphisms of and show that they commute? For (c), isn't this by definition?