TITLE: Geometric interpretation of primitive element theorem?
QUESTION [6 upvotes]: The primitive element theorem is a basic result about field extensions. I was wondering whether there are nice geometric ways to visualize it or think about it. Since field spectra are singletons, it has to be about the non-trivial automorphisms of points (I think), and I don't know how to think about it.

REPLY [10 votes]: Here’s a geometric argument that has nothing to do with algebraic geometry. You may find it insufficiently rigorous, but the idea is certainly sound.
Consider a separable extension $K\supset k$. One consequence of separability is that there are only finitely many intermediate fields $E$, $K\supset E\supset k$. Consider the finitely many proper subfields, and look at their (set-theoretic) union — not their join, not their compositum, just their union. This is a union of finitely many proper $k$-subspaces of $K$, each of them of dimension less than the dimension $n=[K:k]$ of $K$ as a $k$-space. But certainly, if $k$ is infinite, you can’t fill up an $n$-dimensional $k$-space with finitely many subspaces of lower dimension. So there has to be an element $\alpha\in K$ that isn’t in any proper subfield of $K$. Then this $\alpha$ must therefore generate $K$ as a field.