## ‘Is conjugate to over F’ is an equivalence relation

Let $F$ be a field and let $E$ be an algebraic extension of $F$. Prove that the relation $\sim$ defined by $\alpha \sim \beta$ if and only if $\alpha$ and $\beta$ are conjugate over $F$ is an equivalence.

Recall that $\alpha$ and $\beta$ are called conjugate precisely when they have the same minimal polynomial over $F$. We can map $E$ to the set $S$ of all minimal polynomials of elements of $E$ by sending each element to its minimal polynomial; call this map $f$. (Note that $f$ is well-defined since minimal polynomials are unique.) Now $\alpha \sim \beta$ precisely when $f(\alpha) = f(\beta)$. Thus it is clear that $\sim$ is an equivalence relation.