Math 721

Spring 2001

Homework 5

1. Let be a field and an algebraic element of odd degree over Show that

2. Let be an extension of fields with intermediate extensions.

a) Show that is finite if and only if both are finite.

b) Show that if is finite then both divide

c) Show that

d) Show that if are finite and relatively prime, then equality holds in part c.

e) Show that if are algebraic over then so is

3. Compute the Galois group of over , where is the smallest field containing and all the roots of

4. Let be an extension of fields and let be an intermediate ring with identity (that is, ).

a) Show that is an integral domain.

b) Show that the following two conditions are equivalent:

· Every intermediate domain is a field.

· is an algebraic extension of

5. Construct all fields of order 4, 8, and 12 up to isomorphism (if they exist). Prove that your list is complete.