Prove $f$ is irreducible if and only if $f=x-\alpha$ in algebraic closed field

Question

Let ${F}$ be an algebraic closed field and $f(x)\in F[x]$ a monic polynomial.

Prove: $f$ is irreducible if and only if $f=x-\alpha$, for some $\alpha \in F$

So one direction is trivial, but I'm struggling with the second.

I think I need to conclude from that $f$ is irreducible, something about the roots of $f$, but not sure how to work with it.

Any help appreciated.

Answer 1

The point is every polynomial in an algebraically closed field splits into linear factors. This is the definition of algebraically closed. So if $f$ was not of the form $x-\alpha$, then $f$ is of the form $(x-\alpha_1)\dots(x-\alpha_n)$ for some $n \ge 2$. This is clearly not irreducible.