# Fields- again

1. Mar 30, 2005

### 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?

2. Mar 30, 2005

### matt grime

Suppose E/F and K/F are what?

3. Mar 30, 2005

### Palindrom

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

4. Mar 30, 2005

### matt grime

Finite or transcendental?

5. Mar 30, 2005

### Palindrom

It's not mentioned... I guess either of them.

6. Mar 30, 2005

### Palindrom

Help... anyone?

It's to be handed in tomorrow... I really dunno where to start.

7. Mar 30, 2005

### matt grime

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

8. Mar 30, 2005

### 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?

9. Mar 30, 2005

### 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)...

10. Mar 30, 2005

### 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.

11. Mar 31, 2005

### 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.

12. Mar 31, 2005

### 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.

13. Mar 31, 2005

### 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.

14. Mar 31, 2005

### matt grime

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

15. Apr 1, 2005

### 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...

16. Apr 1, 2005

### 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....

17. Apr 1, 2005

### Palindrom

You wanna 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...