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?
On
The proof speaks by itself. But, here some explanation.
Recall that if $f(x)$ is a polynomial in $k[x]$ where $k$ is a field, as $k[x]$ is a Unique factorization domain, there is a unique way to write $f(x)$ in their irreducible factors. So, if we want to find a root of it, it is enough to study one irreducible factor, and we suppose then that $f(x)$ is irreducible. But, this implies that the ideal $(f(x))$ is maximal, so $k[x]/(f(x))$ is a field, that "have a root" for $f(x)$.
The idea behind this, is that in the quotient you will "kill" every term that is a factor of $f(x)$, so $f$ in $x$ will banish. In fact, if we denote $\bar{x}:=x+(f(x))$, we have that $f(X)\in \left(k[x]/(f(x))\right)[X]$ has $x$ as root just as we required.
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 $K[X]/(f)$).