We investigate the role of ultraproducts in proofs in ordinary mathematics, focusing on examples in commutative algebra. We investigate how the properties of an ultraproduct of fields relate to the original fields, and describe a systematic way of taking proofs involving ultraproducts and "unwinding" them into direct proofs. Along the way discover why ultraproducts are harder to remove from some proofs and how these proofs can reveal new field-theoretic notions.