Proving F-Isomorphism Between E and K

  • Thread starter Palindrom
  • Start date
In summary, someone came up with a brilliant idea to try and find a function that wasn't an L1 but whose derivative wasn't.
  • #1
Palindrom
263
0
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?
 
Physics news on Phys.org
  • #2
Suppose E/F and K/F are what?
 
  • #3
matt grime said:
Suppose E/F and K/F are what?
I'm sorry- just assume E and K are extensions of F.
 
  • #4
Finite or transcendental?
 
  • #5
matt grime said:
Finite or transcendental?
It's not mentioned... I guess either of them.
You got any leads?
 
  • #6
Help... anyone?

It's to be handed in tomorrow... I really don't know where to start.
 
  • #7
Try thinking instead of field E(=K) with a subfield (F) that is not preserved by any automorphism of E.
 
  • #8
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
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
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
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
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
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
Oh, no I'm not insulted, and I now think my idea is absolutely crap to boot.
 
  • #15
:smile:
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
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
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...
 

FAQ: Proving F-Isomorphism Between E and K

What is an F-isomorphism between E and K?

An F-isomorphism between E and K is a bijective mapping that preserves the structure of a field F. This means that the mapping preserves the operations of addition, subtraction, multiplication, and division, as well as the identities of 0 and 1.

Why is it important to prove F-isomorphism between E and K?

Proving F-isomorphism between E and K is important because it allows us to show that two fields, E and K, are equivalent in terms of their structure and properties. This is useful in various areas of mathematics and physics, such as algebraic geometry and group theory.

What are the steps involved in proving F-isomorphism between E and K?

The first step is to define the mapping between E and K and show that it is bijective. Then, we need to show that the mapping preserves the operations of addition, subtraction, multiplication, and division. Finally, we need to check that the mapping preserves the identities of 0 and 1.

Are there any properties that an F-isomorphism between E and K must satisfy?

Yes, an F-isomorphism between E and K must satisfy the following properties:

  • It is a bijective mapping
  • It preserves addition: f(a+b) = f(a) + f(b)
  • It preserves subtraction: f(a-b) = f(a) - f(b)
  • It preserves multiplication: f(ab) = f(a) * f(b)
  • It preserves division: f(a/b) = f(a) / f(b)
  • It preserves the identity of 0: f(0) = 0
  • It preserves the identity of 1: f(1) = 1

Can an F-isomorphism between E and K exist if E and K have different characteristics?

No, an F-isomorphism between E and K cannot exist if they have different characteristics. This is because the characteristic of a field is determined by the order of its additive identity, and an F-isomorphism must preserve this property.

Similar threads

I Need clarification on a theorem about field extensions/isomorphisms
Replies
14
Views
2K
I Proving inequality about dimension of quotient vector spaces
Replies
0
Views
833
I Proving properties of polynomial in K[x]
Replies
8
Views
1K
I Finding the orthogonal projection of a vector without an orthogonal basis
Replies
3
Views
2K
I Finite fields, irreducible polynomial and minimal polynomial theorem
Replies
6
Views
2K
I Understanding the concepts of isometric basis and musical isomorphism
Replies
3
Views
2K
MHB The extension KE/F is finite and separable
Replies
1
Views
1K
A Proving the Dual of Schanuel's Lemma
Replies
7
Views
2K
Valuations and places - decomposition and inertia group
Replies
4
Views
1K
MHB The splitting field of f over F is also the splitting field of f also over E
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top