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Jan 15, 2014 at 19:54 comment added KConrad One subgroup of your $G$ is $S_n$, acting on the variables $t_i$ as permutations. The nature of the fixed field of your $K$ by subgroups of $S_n$ is subtle. See a theorem of Swan at en.wikipedia.org/wiki/Rational_variety (note $K$ there is your $\mathbf C$, and $L$ there is your $K$, more or less).
Jan 15, 2014 at 19:25 history asked Julian Rosen CC BY-SA 3.0