$|\bar{F}:F|=\infty$ implies that the degree of the irreducible polynomials in $F[x]$ is unbounded?

Question

Let $F$ be a field and $\bar{F}$ be its algebraic closure. Is it true that $|\bar{F}:F|=\infty$ implies that the degree of the irreducible polynomials in $F[x]$ is unbounded?

I believe it is, and in this case an hint in the right direction for a proof would be enough, but I don't know where to begin

Answer 1

Hint Try contrapositive.

Assume that the degree of irreducible polynomials is bounded. Let $N$ be a bound on this.

Then if $a \in \bar{F}$ we have $$[F(a):F] \leq N$$

Now, pick some $a$ so that $[F(a):F]$ is maximal.

If $b \in \bar{F}$ try to find some $\alpha \in F, \alpha \neq 0$ so that $$a \in F(a+\alpha b)$$

This is always possible if $F$ is infinite.