Let be a field and let be an algebraic extension of . Prove that the relation defined by if and only if and are conjugate over is an equivalence.
Recall that and are called conjugate precisely when they have the same minimal polynomial over . We can map to the set of all minimal polynomials of elements of by sending each element to its minimal polynomial; call this map . (Note that is well-defined since minimal polynomials are unique.) Now precisely when . Thus it is clear that is an equivalence relation.