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If $E$ and $F$ are two fields and $f : E \to F$ and $g : F \to E$ be field homomorphisms. Is it true that $E$ and $F$ are isomorphic?

We know that both $f$ and $g$ are injective which prompts the question.

EDIT: As pointed out in comments and other posts, this is not true in general. Are there any conditions on $E$, $F$ which makes this result true? Algebraically closed, characteristic zero, positive etc.?

Vishal Gupta
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    See http://mathoverflow.net/questions/1058/when-does-cantor-bernstein-hold and http://math.stackexchange.com/questions/257650/analogue-of-the-cantor-bernstein-schroeder-theorem-for-general-algebraic-structu. – lhf Nov 28 '16 at 17:24
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    Noah Snyder in the comments of the SBS blog post linked in the MO thread that lhf links ^ gives the example of $\Bbb C(x)$ and $\Bbb C$. – anon Nov 28 '16 at 17:26
  • @arctictern It wasn't Noah Snyder who gave that example but Scott Carnahan instead, but thanks nonethless. – Vishal Gupta Nov 28 '16 at 17:38
  • Oops, being silly. Sorry Scott. – anon Nov 28 '16 at 17:40
  • no because you always got the trivial homomorphism. – Zelos Malum Nov 28 '16 at 18:49
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    @ZelosMalum There is no trivial homomorphism of fields . . . – Noah Schweber Nov 28 '16 at 21:16

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