Let be a field. Prove that contains a unique smallest subfield and that is isomorphic to either or for some prime . ( is called the prime subfield of .)
We begin with a lemma.
Lemma: Let be a field and let be a set of subfields of . Then is a subfield of . Proof: is a subring of , and contains 1 since for all . Let . Then for each , and so . Then , and thus is a field.
Let be a field, and let denote the set of all subfields of . By the lemma, is a subfield of and is contained in every other subfield.
Now consider the ring homomorphism with . The induced map is an injective ring homomorphism, and by these two previous exercises, is a prime or 0. Certainly for all nonzero , is a unit in . Thus we have an injective homomorphism , where denotes the field of fractions. Note that is a subring of which is a field, and thus . Thus is an isomorphism. Now if , then , and if is prime, then .