Fixed Field of Polynomials in 3 Variables

Asked Aug 27 '22 at 07:30

Active Aug 30 '22 at 19:00

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$K=k(x,y,z)$ be the field of fractions of the integral domain $R=k[x,y,z]$. Hence the field comes equipped with the automorphisms of $G=S_3$ by permuting $x,y$ and $z$.

I want to prove the following

$K^G=k(s_1,s_2,s_3)$
$R^G=k[s_1,s_2,s_3]$
$\mathbb{Z}^G=\mathbb{Z}(s_1,s_2,s_3)$
$K^{A_3}=k(s_1,s_2,s_3,\Delta)$ where $\Delta=(x-y)(y-z)(z-x)$

For part 1, I was able to show using Artin's Theorem but I couldn't find any way to solve $2,3$ and $4$. I was able to prove the one side(the trivial one) inclusion but I need help with the other one(non-trivial)

edited Aug 30 '22 at 19:00

Jyrki Lahtonen

133,153

asked Aug 27 '22 at 07:30

permutation_matrix

1,356

1

In Jacobson's Basic Algebra I (Theorem 2.20 in my edition) there is a proof for the fact that for any commutative ring $R$ we have $R[x_1,x_2,\ldots,x_n]^{S_n}=R[s_1,s_2,\ldots,s_n]$. That takes care of 2 and 3. Number 4 follows from Galois theory by observing that $\Delta^2$ is in the fixed field of $S_3$ while $\Delta$ is not. But only if you assume that the characteristic of $k$ is not equal to two. For otherwise $\Delta$ is also a fixed point. – Jyrki Lahtonen Aug 27 '22 at 07:57
See also Wikipedia. – Jyrki Lahtonen Aug 27 '22 at 07:59
Locally this. – Jyrki Lahtonen Aug 27 '22 at 08:05
@JyrkiLahtonen, Sir there any way to solve it using integral extensions? – permutation_matrix Sep 05 '22 at 10:12

Fixed Field of Polynomials in 3 Variables

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