Is the degree of a field extension inferiorly bounded by the degree of irreducible polynomials?

Question

Let $\overline K$ be the algebraic closure of a field $K$. Let us suppose that there exists a polynomial $P\in K[X]$ that is irreducible over $K$.

Do we have $\deg P\leq[\overline K:K]$?

If so, is there a more general result where we would replace $\overline K$ with an arbitrary extension?