How can the root of an irreducible polynomial exist?

Question

Let $K$ be a field, $P(X)$ an irreducible polynomial in $K[X]$, and let $\alpha$ be a root. This is a very common statement I've seen in lecture notes.

The question is: how does $\alpha$ even exist? If $P(X)$ is an irreducible polynomial, then by definition, shouldn't it have no roots? What exactly is the above statement supposed to mean?

Answer 1

First way: in an algebraically closed extension of $\;K\;$ , such $\;\alpha\;$ exists.

Second way: The ideal $\;\langle P(x)\rangle\le K[x]\;$ is prime and thus maximal in $\;K[x]\;$ , and thus $\;K[x]/\langle P(x)\rangle\;$ is a field. It is now a simple matter to check that $\;\alpha:= x+\langle P(x)\rangle\;$ is a root of $\;P(x)\;$ .