Problem Statement: Let be an integral domain. Show that is a field if and only if has no nontrivial ideals.
Proof: First assume that is a field and let be an ideal of . If then it follows that since for every . Suppose then that and let , then there exists a since by assumption is a field. But since is an ideal it follows that , but since this is only possible if and so it follows that . Thus the only ideals of are the trivial ideals.
Now assume that the only ideals of are and . Fix a non-zero and define by . is a homomorphism with since is an integral domain and so we have no zero divisors. So we have that is one to one and thus is onto since it maps to itself. Now, since it follows that there exists such that . By definition of this implies that and so . Thus every non-zero element in is a unit and so is a field.
Reflection: This technique of using a homomorphism to find an inverse is a really helpful tool. This only works since we can force which is due to the fact that is an integral domain.