TITLE: Does there exist a field which has infinitely many subfields?
QUESTION [9 upvotes]: Does there exist a field which has infinitely many subfields? Does there exist an enormous supply of such fields?

I don't know how to begin.

REPLY [13 votes]: Adding an example of infinitely many intermediate fields of a finite extension $L/K$.
Let be $F$ be an algebraic closure of $\mathbb{F}_2$, so $F$ is infinite. Let $x$ and $y$ be algebraically independent trascendental elements over $F$. Let $L=F(x,y)$ and $K=F(x^2,y^2)$. We easily see that $[L:K]=4$. A basis for this extension consists of $\{1,x,y,xy\}$. Let $\alpha\in F$ be arbitrary. Consider the intermediate space $L_\alpha$ spanned over $K$ by $\{1,x+\alpha y\}$. We have 
$$
L_\alpha\cap L_{\alpha'}=K
$$
whenever $\alpha\neq\alpha'$ (linear algebra), so the spaces $L_\alpha$ are distinct. The key is that $L_\alpha$ is also a subfield. This is because
$$
(x+\alpha y)^2=x^2+2\alpha xy+\alpha^2y^2=x^2+\alpha^2y^2\in K.
$$
This does not contradict the argument in Pete L. Clark's answer, because the extension $L/K$ is purely inseparable, and thus cannot be contained in a Galois extension. And that was the key in his argument in the case of number fields.