I have a question on field extensions, and I can't seem to find precise answers when browsing through online notes etc. Here it is: suppose $K$ and $k$ are fields with $k \leq K$ and $[K : k] = m$ is finite. Without adding extra assumptions on $m$, when can we conclude that there is an element $f(x) \in k[x]$ of degree $m$ such that $k[x]/(f(x)) \cong K$ ? What if we replace "of degree $m$" by "of degree divisible by $m$" ?
Thanks !
This is the case iff $K/k$ is a simple field extension. It is known that finite separable extensions are simple (Primitive Element Theorem), but there are finite extensions which are not simple, for example $\mathbb{F}_p(x,y)$ over $\mathbb{F}_p(x^p,y^p)$.