if a field F has order n, then F* has order n-1

5. Suppose E is a finite extension of a field F and K and L are subfields of E that contain F and are normal over F. Prove that KnL is normal over F.

9. Recall from Section 2 that a field F is called an ordered field if there exists a subset P of F (called the set of positive elements) such that (a) sums and products of elements in P are in P, and (b) for each element a in F, one and only one of the following possibilities holds: a e P, a = Prove that the field of complex numbers is not an ordered field. 0, - a e P.

If K is a subfield of the Galois field GF(p\power{n}), then there exists an integer m such that K contains p\power{m} elements and m is a divisor of n.