Constructing Field Extensions from Irreducible Polynomials Suppose f(x) $\in$ F[x] is a non-zero irreducible polynomial of degree n < $\infty$ over a field F, then E = F[x]/(f) is a field extension E\F of degree n. 
My question is: Can every field extension of F that is of finite degree be constructed in this way?

Exactly the simple finite field extensions arise this way. Every separable finite field extension is simple (see the Primitive element Theorem). An example of a finite field extension which fails to be simple is $\mathbb{F}_p(x^p,y^p) \to \mathbb{F}_p(x,y)$. In fact, it has degree $p^2$, but every element has degree $\leq p$.
