Geodesic
Note on a variation of the Schröder-Bernstein problem for fields
Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 717-720.

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In this note we study fields $F$ with the property that the simple transcendental extension $F(u)$ of $F$ is isomorphic to some subfield of $F$ but not isomorphic to $F$. Such a field provides one type of solution of the Schröder-Bernstein problem for fields.
Classification : 12E99, 12F05, 12F20
Keywords: field; subfield; isomorphism; transcendental extension; algebraic extension 
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     author = {Cater, F. S.},
     title = {Note on a variation of the {Schr\"oder-Bernstein} problem for fields},
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Cater, F. S. Note on a variation of the Schröder-Bernstein problem for fields. Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 717-720. http://geodesic.mathdoc.fr/item/CMJ_2002__52_4_a3/