Given an arbitrary algebraically closed field F, can we always construct a 'superfield' K such that F<K and K is also algebraically closed? If so, how?
I'm curious whether you can always extend an already closed field to get a 'bigger' field that contains it (as a distinct subset).
An example of two fields with this property would be the algebraic numbers and the complex numbers. Both are algebraically closed, and yet the complex numbers extend the algebraic numbers.