Let $\mathbb F_2$ be a field of order $2$. Let $F$ be the function field $\mathbb F_2(x,y)$, i.e., $x,y$ are independent variables and $F$ is the field of quotients of the polynomial rings $\mathbb F_2[x,y]$. Find a finite extension $L$ of $F$ such that it has infinitely many intermediate subfields containing $F$.

I have tried a lot but cannot give an example. Thank you a lot.

  • $\begingroup$ Please show some effort. $\endgroup$
    – LASV
    Dec 11, 2013 at 4:56

1 Answer 1


Take $L=\mathbb F_2(\sqrt x,\sqrt y)$.

  • 5
    $\begingroup$ how to construct the intermediate fields? $\endgroup$
    – Shiquan
    Dec 13, 2013 at 2:16

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