Math 721
Spring 2001
Homework 5
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
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.