Lagrange's rational function theorem states that if one has two rational functions in multiple varia

enfocarteu7z

enfocarteu7z

Answered question

2022-02-16

Lagrange's rational function theorem states that if one has two rational functions in multiple variables f(x1,x2,xn) and g(x1,x2,xn) 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.

Answer & Explanation

shotokan0758s

shotokan0758s

Beginner2022-02-17Added 8 answers

Consider the field K=Q(x1,x2,xn), and consider KgK to be the subfield generated by g. Define
Hg=Aut(KKg={σϵAut(K):σ(α)=ααϵKg}
Similarly, define Kf and Hf. Then you want to show that
KfKgHgHf
If all the hypotheses are satisfied, this is merely the Fundamental Theorem of Galois Theory

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