Related papers: Noether's Problem on Semidirect Product Groups

Let K be any field and G be a finite group. We will prove that, if K is any field, p an odd prime number, and G is a non-abelian group of exponent p with |G|=p^3 or p^4 satisfying [K(\zeta_p):K] <= 2, then K(G) is rational over K. We will…

Motivated by the classical Noether's problem, J. Alev and F. Dumas proposed the following question, commonly referred to as the noncommutative Noether's problem: Let a finite group $G$ act linearly on $\mathbb{C}^n,$ inducing the action on…

Let $k$ be any field, $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $h\cdot x_g=x_{hg}$ for any $h,g\in G$. Define $k(G)=k(x_g : g\in G)^G$. Noether's problem asks whether $k(G)$ is rational (= purely…

Rationality problems of algebraic k-tori are closely related to rationality problems of the invariant field, also known as Noether's Problem. We describe how a function field of algebraic k-tori can be identified as an invariant field under…

Let $K$ be a field of characteristic not two and $K(x,y,z)$ the rational function field over $K$ with three variables $x,y,z$. Let $G$ be a finite group of acting on $K(x,y,z)$ by monomial $K$-automorphisms. We consider the rationality…

We address the Noncommutative Noether's Problem on the invariants of Weyl fields for linear actions of finite groups. We prove that if the variety An(k)/G is rational then the Noncommutative Noether's Problem is positively solved for G and…

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 an infinite field. The notion of retract $k$-rationality was introduced by Saltman in the study of Noether's problem and other rationality problems. We will investigate the retract rationality of a field in this paper. Theorem 1.…

In 1918, Noether published a paper where she studied such a problem, now called Noether's problem on rationality: Let $L=K\left( t_{1},t_{2},\cdots ,t_{n}\right) $ be a purely transcendental extension over a field $K$ and $G$ a finite…

Let $G$ be a finite group acting on the rational function field $\mathbb{C}(x_g : g\in G)$ by $\mathbb{C}$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the invariant field $\mathbb{C}(G)=k(x_g : g\in…

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…

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 ${\rm PGL}_2$…

Noether's problem asks whether, for a given field K and finite group G, the fixed field L := K(x_h : h \in G)^G is a purely transcendental extension of K, where G acts on the x_h by gx_h = x_gh. The field L is naturally the function field…

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 finite subgroup of $\mathrm{Aut}_k(K(x_1, \ldots, x_n))$ where $K/k$ is a finite field extension and $K(x_1,\ldots,x_n)$ is the rational function field with $n$ variables over $K$. The action of $G$ on $K(x_1, \ldots, x_n)$ is…

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 $k(x_1,x_2,\ldots,x_n)$ via $k$-automorphisms defined by $\sigma\cdot x_i=x_{\sigma(i)}$ for any…

If $G$ is a finite $\ell$-group acting on an affine space $\mathbb{A}^n$ over a finite field $K$ of cardinality prime to $\ell$, Serre has shown that there exists a rational fixed point. We generalize this to the case where $K$ is a…

Let $K=\mathbb{F}_q(C)$ be the global function field of rational functions over a smooth and projective curve $C$ defined over a finite field $\mathbb{F}_q$. The ring of regular functions on $C-S$ where $S \neq \emptyset$ is any finite set…

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…

Let G be a finite group and let F be a field of characteristic zero. In this paper we construct a generic G-crossed product over F using generic graded matrices. The center of this generic G-crossed product, denoted by F(G), is then the…