Proving F-Isomorphism Between E and K

[Palindrom]

O.K.
Here it is:
Prove or find a counter example.
Suppose E/F, and K/F. Then E~K (iso.) => E and K are F-isomorphic.

I can prove it for F=Q or any finite field.
Is it true in general?

 

[matt grime]

Suppose E/F and K/F are what?

 

[Palindrom]

Suppose E/F and K/F are what?

I'm sorry- just assume E and K are extensions of F.

 

[matt grime]

Finite or transcendental?

 

[Palindrom]

Finite or transcendental?

It's not mentioned... I guess either of them.
You got any leads?

 

[Palindrom]

Help... anyone?

It's to be handed in tomorrow... I really don't know where to start.

 

[matt grime]

Try thinking instead of field E(=K) with a subfield (F) that is not preserved by any automorphism of E.

 

[Palindrom]

So you're going for the counter example?

I tried your idea, but I couldn't find an example.
(I really did :) )

Are you sure it's wrong?

 

[Palindrom]

btw, it wouldn't be good- because E is still F-isomorphic to itself.

I would need 2 fields that aren't F-iso. (and of course, I'd have to prove they aren't)...

 

[matt grime]

All I said was, sicne E and K are isomorphic, that we may replace K with E. That is is there a Field, which cotains a subfield (over which it is an extension), such that no isomorphism is an F-isomorphism. I really have just restated the question: any F-isomorphism is still an isomorphism.

I don't know whether the result is true to be honest.

 

[Palindrom]

Well, I went to see my Professor today- he said it wasn't true.
Told me to keep thinking about a counter example, and that it's quite untrivial. He promised to answer me next week though.

I must have not understood what you meant, by the way- didn't you suggest that I would find an extension E of F that has no F automorphisms? Because that's what I thought you said, I apologize if I got you wrong.

 

[matt grime]

No I said to find two objects that are isomprohic, ie we may as well replace them the same symbol and that contain F as (sub)field over which they are extensions such that no automorphism preserves F. I had a feeling there would be a counter example, and I feel that I ought to be able to come up with one, but I've not spent long enough on it, and, if you don't mind, dont' really intend to try figuring it out.

 

[Palindrom]

I'm not asking you to, if you don't want to.
I feel you might be a bit insulted- if you are, it is absolutely not my intention.

 

[matt grime]

Oh, no I'm not insulted, and I now think my idea is absolutely crap to boot.

 

[Palindrom]

Then you must know how I feel... I'll keep trying though, if I finish up all the other weird stuff I have to do.
Have a nice weekend...

 

[matt grime]

Oh, I have one stupid idea an hour or its a slow day. SOmetimes I sadly tell other people of the stupid idea before I figure out its stupid. And they pay me to do this...

 

[Palindrom]

You want to hear stupid?

About 3 or 4 friends of mine thought for about 3 days about the next problem: Find a function that isn't L1 but whose derivative is.

Then one of them got a brilliant idea: take f(x)=const.

And you should have seen what they where trying to do before that idea- I heard the words "delta function" quite a few times that week...