What are the simplest examples of situations where in a field $F$ there are two subfields $L_1$ and $L_2$ such that extensions $F/L_1$ and $F/L_2$ are finite, degrees are different $$ [F:L_1] \neq [F:L_2], $$ but fields $L_1$ and $L_2$ are isomorphic as abstract fields.
2026-04-02 14:08:30.1775138910
Fields extensions over isomorphic fields of different degrees
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For any field $k$, and an indeterminate $t$ over $k$, take $F = k(t)$, $L_1 = F$, $L_2 = k(t^2)$. Then $[F : L_1] = 1 \ne 2 = [F : L_2]$, but as abstract fields, $L_1 \cong L_2$ (as $L_2$ is the field of fractions of $k[t^2]$, which is abstractly isomorphic to a polynomial ring in $1$ variable over $k$).