# Non-algebraic Galois Extension

• 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?
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!)