It is well known that if we want to perform an extension of a field, say $\mathbb{F}$, we can extend it via $\frac{\mathbb{F}[X]}{\langle f(x)\rangle}$ where $f(x)$ is an irreducible polynomial over $\mathbb{F}[X].$
Now I am curious, suppose we have a degree $n$ field extension of $\mathbb{F}$, can it always be realised as $\mathbb{F}[X]/\langle g(x)\rangle$ for some $g(x)$, irreducible and degree $n$? Put differently, is there 'a converse' to my first paragraph?
I hope what I asked made sense, much appreciation for any hint in advance!