Related papers: On a rationality problem for fields of cross-ratio…

In this paper, we give a brief survey of recent developments on Noether's problem and rationality problem for multiplicative invariant fields including author's recent papers Hoshi [Hos15] about Noether's problem over Q, Hoshi, Kang and…

We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ of $K/k$ whose Galois closures $L/k$ are $A_5\simeq {\rm PSL}_2(\mathbb{F}_4)$ and ${\rm…

We generalize a result of F.\ Legrand about the existence of non-parametric Galois extensions for a given group $G$. More precisely, for a $K$-regular Galois extension $F|K(t)$, we consider the translates $F(s)|K(s)$ by an extension…

Let $p$ be an odd prime and $G$ be a nonabelian group of order $p^{n}$ with the presentation $$<\alpha,\beta,\gamma\mid \alpha^{p^{a}}=\beta^{p^{b}}=\gamma^{p^{c}}=1,…

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 $\sigma \in G$,…

In this article, we construct a generating set of rational invariants for the action of the orthogonal group $\text{O}(n)$ on the space $\mathbb{R}[x_1,\dots,x_n]_{2d}$ of real homogeneous polynomials of even degree $2d$. This generalizes a…

Ideas and techniques from Khare's and Wintenberger's article on the proof of Serre's conjecture for odd conductors are used to establish that for a fixed prime l infinitely many of the groups PSL_2(F_{l^r}) (for r running) occur as Galois…

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 $\sigma\in G$,…

We give a positive solution to Noether's rationality problem for certain index $p$ subgroups of the $p$-Sylow subgoups of symmetric groups.

We consider the Noether Problem for stable and retract rationality for the sequence of $d$-torsion subgroups $T[d]$ of a torus $T$, $d\geq 1$. We show that the answer to these questions only depends on $d\pmod{e(T)}$, where $e(T)$ is the…

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…

Given a generic field extension F/k of degree n>3 (i.e. the Galois group of the normal closure of F is isomorphic to the symmetric group $S_n$), we prove that the norm torus, defined as the kernel of the norm map $N:R_{F/k}(G_m)\to\G_m$, is…

Let $G$ be a finite group. The aim of this paper is to study the number of solutions $S\subseteq G$ of the equation $\mho^{\{n\}}(S)=L$, where $L$ is a non-empty subset of $G$, $n$ is a positive integer and $\mho^{\{n\}}(S)=\{ s^n \ | \…

The unique irreducible representation of $\SL_2(\R)$ on $\R^n$ induces an action, called the \textit{linear action}, of $\SL_2(\Z)$ on the torus $\T^n$ for every $n\geq 2$. For $n$ odd, it factors through $\PSL_2(\Z)$, so we denote by $G_n$…

This is the first of a series of papers, where we investigate hierarchies of generalized {L}\"{u}roth problems on the hierarchy of rationality, starting with the obvious hierarchy between the rationality and the ruledness. Our primary goal…

We study whether the norm one torus associated with a finite separable non-Galois field extension $K/k$ is $p$-retract rational over $k$ for a prime $p$, focusing on the case where the Galois group of the Galois closure of $K/k$ is either…

Given a field K, one may ask which finite groups are Galois groups of field extensions L/K such that L is a maximal subfield of a division algebra with center K. This connection between inverse Galois theory and division algebras was first…

Rationality results are proved for the ratios of critical values of Rankin-Selberg L-functions of GL(n) x GL(n') over a totally imaginary field F, by studying rank-one Eisenstein cohomology for the group GL(N)/F, where N = n+n',…

If the $\ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For…

Which subgroups of the symmetric group S_n arise as invariance groups of n-variable functions defined on a k-element domain? It appears that the higher the difference n-k, the more difficult it is to answer this question. For k>=n, the…