Non-algebraic Galois Extension

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May 27th 2008, 07:20 PM
ThePerfectHacker

Non-algebraic Galois Extension

Say $E/F$ is Galois*.
Must it be the case that $E/F$ is algebraic?
(This was not addressed in my book).

*)I just realized how bad my question is. A Galois extension $E/F$ is an algebraic extension such that $F = E^{\text{Gal}(E/F)}$ . Note we use "algebraic" within the definition itself. My question would be more appropriately asked if $F = E^{\text{Gal}(E/F)}$ then must it follow that $E/F$ is algebraic?
May 28th 2008, 07:09 PM
NonCommAlg

Quote:

Originally Posted by ThePerfectHacker

My question would be more appropriately asked if $F = E^{\text{Gal}(E/F)}$ then must it follow that $E/F$ is algebraic?

this is a good question! the answer is No! because, for example, if $x$ is transcendental over $F$ and $E=F(x)$ , then

it can be proved (not very easily though!) that $E^{\text{Gal}(E/F)}=F.$ the main part of the proof is to show that $\text{Gal}(E/F)$

consists of all maps $\sigma$ defined by $\sigma(f(x))=f(\frac{ax+b}{cx+d}), \ \forall f \in E,$ where $a,b,c,d \in F$ and $ad - bc \neq 0.$
May 28th 2008, 07:18 PM
ThePerfectHacker

Quote:

Originally Posted by NonCommAlg

this is a good question! the answer is No! because, for example, if $x$ is transcendental over $F$ and $E=F(x)$ , then

it can be proved (not very easily though!) that $E^{\text{Gal}(E/F)}=F.$ the main part of the proof is to show that $\text{Gal}(E/F)$

consists of all maps $\sigma$ defined by $\sigma(f(x))=f(\frac{ax+b}{cx+d}), \ \forall f \in E,$ where $a,b,c,d \in F$ and $ad - bc \neq 0.$

I actually was thinking about $\mathbb{Q}(\pi)/\mathbb{Q}$ , 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).
May 28th 2008, 07:56 PM
NonCommAlg

Quote:

Originally Posted by ThePerfectHacker

Now this leads me to my next question: is there anything interesting about a non-algebraic Galois extension?

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 $E^{\text{Gal}(E/K)}=K$ , 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!)