Prove that any subfield of must contain .
By the previous exercise, contains a unique inclusion-smallest subfield which is isomorphic either to for a prime or to .
Suppose the unique smallest subfield of is isomorphic to , and let in this subfield be nonzero. Then in , and since is a unit, , a contradiction.
Thus the unique smallest subfield of is isomorphic to . In particular, any subfield of contains a subfield isomorphic to .