This is from Serge Lang's Algebra, in Chapter 5.
If $K$ is a field, and $K\subset E$ is a field extension, then for $\alpha\in E$, then $K(\alpha)$ is defined to be the smallest subfield of $E$ which contains $K$ and $\alpha$, which is given by the field of quotients $K(\alpha) = \left\{\frac{f(\alpha)}{g(\alpha)}:f(x),g(x)\in K[x]\right\}$.
I tried to prove this remark but am stuck on $(\subset)$.
The right hand side clearly contains $K$ by taking $f(x) = k$, and $g(x) = 1$ for any $k\in K$. But how can I represent $\alpha$ as such a quotient?
Edit: I think (my memory is a bit rusty here) that if $K$ is a field and $\alpha$ is some object not in $K$, we construct the field $K[x]$ which contains $K$ as a subfield, and identify $\alpha$ with the indeterminant $x$. Do I have to examine this embedding more closely or is this just going down the wrong train of thought?