There is a theorem in my book which says the following:
Let $K$ be a field and let $f(X) \in K[X]$ be irreducible over $K$. Then there exists a field extension $L/K$, such that $\exists u \in L$: $f(u) = 0$.
Proof: Notice that $K\subseteq K[X]/(f(x))$. Clearly, $K[X]/(f(x))$ is a field since $(f(x))$ is maximum ideal. If we take $u = \overline{X}$, then $f(u) = f(\overline{X})= \overline{f(X)} = 0$.
Question: I really don't get if $u = ...$ in the proof, since I don't know what $\overline{X}$ means in this context, does anyone know?
Thanks in advance.
Unfortunately I cannot provide a link to the book as it is my dutch (non pdf) syllabus for my algebra course.
EDIT: Seems like I get it now: $f(\overline{X}) = \overline{f(X)} = f(x)+ (f(x)) = (f(x))$ which is $0$ for $K\subseteq K[X]/(f(x))$.