Say is Galois*.
Must it be the case that is algebraic?
(This was not addressed in my book).
*)I just realized how bad my question is. A Galois extension is an algebraic extension such that . Note we use "algebraic" within the definition itself. My question would be more appropriately asked if then must it follow that is algebraic?
I actually was thinking about , but I never did any of the details to prove it is Galois.
Now this leads me to my next question: is there anything interesting about a non-algebraic Galois extension?
(Maybe my books talk about this stuff near the end on transcendental extensions,
but it would be interesting to me to know ahead of time).
i'm not sure, probably it's more complicated than interesting! for example we know that in algebraic Galois extensions,
if E/F is Galois and K is a subfield of E which contains F, then E/K would be Galois too. now in order to have this nice
property in non-algebraic extensions we may define that E/F to be Galois if , for any subfield K of E which
contains F. but then we'd have a problem: with this definition there is no non-algebraic Galois extension of any field of
characteristic p > 0 (this is easy to prove!)