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Specific question: Let $F$ be a field and assume that $\mathbb{Q}$ is a proper subfield of $F$. Can $F$ be isomorphic to $\mathbb{Q}$?


Studying the foundaments of field theory I have to ask: Can a field be isomorphic to one of its proper subfields?

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    $\begingroup$ See here for an example - namely, that the algebraic closure of $\Bbb C(x)$ is isomorphic to $\Bbb C$. $\endgroup$ – Stahl Jan 10 '14 at 7:15
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Yes, given any field $k$, $k(x)$ is isomorphic to $k(x^2)$.

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