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If $E \subseteq F=k(x_1,\ldots,x_r)$, satisfies $E(x_1^{i_1},\ldots,x_r^{i_r})=F$, for every $(i_1,\ldots,i_r) \neq (0,\ldots,0)$, then $[F:E] \leq 2$
For $r \geq 2$, let $A_r=\mathbb{C}[x_1,\ldots,x_r]$,
$F_r=\mathbb{C}(x_1,\ldots,x_r)$ the field of fractions of $A_r$, and $E_r \subseteq F_r$ an arbitrary subfield of $F_r$ with $[F_r:E_r] < \...
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