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Field extensions

  1. Apr 18, 2005 #1
    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.
  2. jcsd
  3. Apr 18, 2005 #2


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    thats obviously false. look at any example at all.

    just extend your proof from field extensions, to fields.
  4. Apr 18, 2005 #3
    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....
  5. Apr 18, 2005 #4
    Ok here's a counterexample. Take the two field extension of the rationals [tex]\mathbb{Q}(i)[/tex] and [tex]\mathbb{Q}(\sqrt{3}i) [/tex]. Both are of degree 2. But then there is an isomorphism [tex] f(0) = f(i^2 +1) = (f(i))^2 +f(1) = (f(i))^2 +1[/tex] But there are no elements in [tex]\mathbb{Q}(\sqrt{3}i) [/tex] that satisfies this.
  6. Apr 18, 2005 #5
    Nice....that's wonderful. Thanks snoble! Exploit the isomorphism property, i get it. :smile:
  7. Apr 18, 2005 #6


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    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?
    Last edited: Apr 19, 2005
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