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.

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Lagrange&#039;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.

shotokan0758s · Accepted Answer

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

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

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