Within this entry, denotes the set of all squarefree integers not equal to .

Theorem   Let with . Then and are not isomorphic.

Proof. Suppose that and are isomorphic. Let be a field isomorphism. Recall that field homomorphisms fix prime subfields. Thus, for every , .

Let with . Since and is injective, . Also, . If , then , a contradiction. Thus, . Therefore, . Since is squarefree, . Hence, , a contradiction. It follows that and are not isomorphic.

This yields an obvious corollary:

Proof. Note that there are infinitely many elements of . Moreover, if and are distinct elements of , then and are not isomorphic and thus cannot be equal.

Note that the above corollary could have also been obtained by using the result regarding Galois groups of finite abelian extensions of . On the other hand, using this result to prove the above corollary can be likened to “using a sledgehammer to kill a housefly”.