Algebraically closed fields with proper maximal subfields  Is there a classification of the algebraically closed fields that have maximal proper subfields ? 
And if an algebraically closed field has a maximal proper subfield, is that subfield unique ? 
Summarizing the answers, an algebraically closed field has a maximal subfield if and only if its characteristic is zero and such a maximal subfield is never unique. 
 A: This occurs iff the field has characteristic 0.  By KConrad's comment, being characteristic 0 is certainly a necessary condition.  Conversely, given an algebraically closed field K of characteristic 0, we can use Zorn's Lemma to find a maximal ordered subfield F.  Since K is algebraically closed, F must be real closed.  But also K must be algebraic over F or else we could pick a transcendental element and adjoin it to F (make it infinitely larger than all elements of F, i.e., use lexicographic ordering).  Hence K must be of degree 2 over F and thus F is a maximal proper subfield.
Also by KConrad's comment, this is never unique; just apply an automorphism of K that takes, e.g., $\sqrt[3]{2}$ to $\omega \sqrt[3]{2}$, where $\omega$ is a primitive cube root of unity.
A: If $F$ is a maximal proper subfield of a field $K$, then $K=F(x)$ for any $x\in K\setminus F$. Next, $x$ must be algebraic over $F$ (otherwise $F\subsetneq F(x^2)\subsetneq F(x)\subset K$). So $K$ is finite over $F$, and if $K$ is algebraically closed it is well known (cf. KConrad's comment) that $F$ is a real closed field and $K=F(\sqrt{-1})$.
