Motivation for the method of adjoining roots of polynomials In Galois theory we learned the standard method of adjoining a root of an irreducible polynomial. More precisely, we saw that if $K$ is a field and $f\in K[x]$ is irreducible then the field $K[x]/(f)$ contains a root of $f$ (namely $x+(f)$). I understand the statement and proof of the theorem completely. But I am interested in the motivation behind this method as it seems very abstract and unintuitive even though it works. Could someone please explain what motivated this idea?

We observe that $K$ has no root of $f$, so we just let $x$ be a root (this corresponds to looking at $K[X]$). Well if $x$ needs to be a root then $f(x)$ better be $0$, so let's just quotient by $f(x)$ in order to make $f(x)=0$ (this corresponds to looking at $K[X]/(f)$).