相关论文: Rational invariants for subgroups of S_5 and S_7

Let $G$ be a subgroup of $S_6$, the symmetric group of degree 6. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,...,x_6)$ via $k$-automorphisms defined by $\sigma\cdot x_i=x_{\sigma(i)}$ for any $\sigma\in G$,…

Let $G$ be a subgroup of $S_{n}$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_{1},\cdots,x_{n})$ via $k$-automorphisms defined by $\sigma\cdot x_{i}:=x_{\sigma\cdot i}$ for…

For a transitive subgroup $G \le S_6$ which contain $C_3 \times C_3$ as subgroup, we prove that $K(x_1,\dots,x_6)^G$ is rational over $K$, where $K$ is any field, and $G$ acts naturally on $K(x_1,\dots,x_6)$ by permutations on the…

For any field K and any transitive subgroup G of S_8, let G acts naturally on K(x_1, . . ., x_8) by permutations of the variables, we prove that under some minor conditions K(x_1, . . ., x_8)^G is always K-rational except G is A_8 or G is…

Let $k$ be a field, $x_1, \dots, x_n$ be independent variables and $L_n = k(x_1, \dots, x_n)$. The symmetric group $\Sigma_n$ acts on $L_n$ by permuting the variables, and the projective linear group $\text{PGL}_2$ acts by \[…

Let $G$ be a finite group acting on $k(x_1,...,x_n)$, the rational function field of $n$ variables over a field $k$. The action is called a purely monomial action if $\sigma...x_j=\prod_{1\le i\le n} x_i^{a_{ij}}$ for all $\sigma \in G$,…

Let $k$ be a field, $G$ be a finite group and $k(x_g:g\in G)$ be the rational function field over $k$, on which $G$ acts by $k$-automorphisms defined by $h\cdot x_g=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the fixed…

Let $G$ be a finite 2-group and $K$ be a field satisfying that (i) $\fn{char}K\ne 2$, and (ii) $\sqrt{a}\in K$ for any $a\in K$. If $G$ acts on the rational function field $K(x,y,z)$ by monomial $K$-automorphisms, then the fixed field…

Let $G$ be a finite group, $k$ be a field and $G\to GL(V_{\rm reg})$ be the regular representation of $G$ over $k$. Then $G$ acts naturally on the rational function field $k(V_{\rm reg})$ by $k$-automorphisms. Define $k(G)$ to be the fixed…

Let $G$ be a finite group and $k$ be a field. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $g\cdot x_h=x_{gh}$ for any $g,h\in G$. Noether's problem asks whether the fixed field $k(G)=k(x_g:g\in…

Let $G$ be a finite group and $k$ be a field. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $g\cdot x_h=x_{gh}$ for any $g,h\in G$. Noether's problem asks whether the fixed field $k(G)=k(x_g:g\in…

Let $k$ be a field, $G$ be a finite group, $k(x(g):g\in G)$ be the rational function field with the variables $x(g)$ where $g\in G$. The group $G$ acts on $k(x(g):g\in G)$ by $k$-automorphisms where $h\cdot x(g)=x(hg)$ for all $h,g\in G$.…

Let $G$ be a finite subgroup of $GL_4(\bm{Q})$. The group $G$ induces an action on $\bm{Q}(x_1,x_2,x_3,x_4)$, the rational function field of four variables over $\bm{Q}$. Theorem. The fixed subfield…

This is a survey on the ancient question : Let G be a reductive group over an algebraically closed field k and let V be a vector space over k with an almost free linear action of G on V. Let k(V) denote the field of rational functions on V.…

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism F, we denote by k(X)^F its field of invariants, i.e. the set of rational functions f on X such that f(F)=f. Let n(F)…

Let $K$ be any field and $G$ be a finite group. Let $G$ act on the rational function field $K(x_g: \ g \in G)$ by $K$-automorphisms defined by $g \cdot x_h=x_{gh}$ for any $g, \ h \in G$. Denote by $K(G)$ the fixed field $K(x_g: \ g \in…

Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's…

Let $k$ be a field, $n \geqslant 5$ be an integer, $x_1, \dots, x_n$ be independent variables and $L_n = k(x_1, \dots, x_n)$. The symmetric group $S_n$ acts on $L_n$ by permuting the variables, and the projective linear group ${\rm PGL}_2$…

Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n-transitively on X, for all natural integers n. As an application…

Let ${\mathbb F}_q$ be a finite field of characteristic two and ${\mathbb F}_q(X_1,...,X_n)$ a rational function field. We use matrix methods to obtain explicit transcendental bases of the invariant subfields of orthogonal groups and…