enfocarteu7z

2022-02-16

Lagrange's rational function theorem states that if one has two rational functions in multiple variables $f\left({x}_{1},{x}_{2},\dots {x}_{n}\right)$ and $g\left({x}_{1},{x}_{2},\dots {x}_{n}\right)$ then one can express f as a rational function in g if and only if the set of permutations that keep g unchanged is a subset of the set of permutations that preserve f.
Is anyone familiar with the proof of this theorem? While it is fairly clear that if f can be expressed in terms of g the set of permutations that keep g unchanged has to be the subset of those that keep f unchanged, the converse is far from obvious.

shotokan0758s

Consider the field $K=\mathbb{Q}\left({x}_{1},{x}_{2},\dots {x}_{n}\right)$, and consider ${K}_{g}\subset K$ to be the subfield generated by g. Define
${H}_{g}=Aut\left(\frac{K}{{K}_{g}}=\left\{\sigma ϵAut\left(K\right):\sigma \left(\alpha \right)=\alpha \mathrm{\forall }\alpha ϵ{K}_{g}\right\}$
Similarly, define ${K}_{f}$ and ${H}_{f}$. Then you want to show that
${K}_{f}\subset {K}_{g}⇔{H}_{g}\subset {H}_{f}$
If all the hypotheses are satisfied, this is merely the Fundamental Theorem of Galois Theory

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