# Prove/Disprove: F1 and F2 Isomorphic as Fields

• T-O7
In summary, the conversation discusses the statement that if F1 and F2 are two finite field extensions of a field K with equal degrees, then they are isomorphic as fields. The speaker initially mentions that the statement is false if "isomorphic as fields" is replaced with "isomorphic as field extensions" and asks for help in proving or disproving the statement. Another person suggests using a counterexample to prove it false, which leads to an example involving two field extensions of the rationals. The speaker then expresses understanding and thanks the person for their help. The conversation ends with a hint to consider a base field and a suggestion to try the same problem with finite fields.
T-O7
Hey all,
I need to prove (or disprove) the following statement:

F1 and F2 are two finite field extensions of a field K. Assume [F1:K]=[F2:K]. Then F1 and F2 are isomorphic as fields.

Some help would be much appreciated.
I know the statement is false if i replace "isomorphic as fields" by "isomorphic as field extensions", but that's all i can think of so far.

thats obviously false. look at any example at all.

just extend your proof from field extensions, to fields.

Sorry...i don't know what you mean.
I can come up with counter examples if it said "isomorphic as field extensions", because then i can just use the result that F(a) and F(b) are F-isomorphic iff a and b have the same irreducible poly.
But I'm stuck in this case, with general field isomorphisms...

Ok here's a counterexample. Take the two field extension of the rationals $$\mathbb{Q}(i)$$ and $$\mathbb{Q}(\sqrt{3}i)$$. Both are of degree 2. But then there is an isomorphism $$f(0) = f(i^2 +1) = (f(i))^2 +f(1) = (f(i))^2 +1$$ But there are no elements in $$\mathbb{Q}(\sqrt{3}i)$$ that satisfies this.

Nice...that's wonderful. Thanks snoble! Exploit the isomorphism property, i get it.

my hint was meant to get you to consider a base field like Q where every isomorphism is the identity, then your same proof works as for extensions. see?

by the way try the same problem for finite fields. i.e. suppose two finite fields have the same number of elements, what then?

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## 1. What does it mean for two fields to be isomorphic?

Two fields F1 and F2 are isomorphic if there exists a bijective homomorphism between them, which means that there is a one-to-one mapping between the elements of F1 and F2 that preserves the operations of addition and multiplication.

## 2. How can we prove that two fields are isomorphic?

To prove that two fields F1 and F2 are isomorphic, we need to show that there exists a homomorphism from F1 to F2 and from F2 to F1. This can be done by defining a mapping between the elements of F1 and F2 and showing that it preserves the operations of addition and multiplication.

## 3. Can two fields be isomorphic if they have different numbers of elements?

No, two fields cannot be isomorphic if they have different numbers of elements. Isomorphism requires a one-to-one mapping between the elements of the two fields, which is not possible if the fields have different numbers of elements.

## 4. Is isomorphism a symmetric relation?

Yes, isomorphism is a symmetric relation. This means that if two fields F1 and F2 are isomorphic, then F2 is also isomorphic to F1. This is because the bijective homomorphism between the two fields works in both directions.

## 5. Can two fields with different characteristics be isomorphic?

No, two fields with different characteristics cannot be isomorphic. The characteristic of a field is the smallest positive integer n such that n multiplied by any element of the field equals 0. This characteristic must be the same for two fields to be isomorphic.

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