# On Noether's problem for cyclic groups of prime order

@article{Hoshi2014OnNP, title={On Noether's problem for cyclic groups of prime order}, author={Akinari Hoshi}, journal={arXiv: Number Theory}, year={2014} }

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g\,|\,g\in G)$ by $k$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the invariant field $k(G)=k(x_g\,|\,g\in G)^G$ is rational (i.e. purely transcendental) over $k$. In 1974, Lenstra gave a necessary and sufficient condition to this problem for abelian groups $G$. However, even for the cyclic group $C_p$ of prime order $p$, it is unknown whether there exist infinitely many… Expand

#### 8 Citations

Birational classification of fields of invariants for groups of order $128$

- Mathematics
- 2014

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… Expand

On Noether's rationality problem for cyclic groups over $\mathbb{Q}$

- Mathematics
- 2016

Let $p$ be a prime number. Let $C_p$, the cyclic group of order $p$, permute transitively a set of indeterminates $\{ x_1,\ldots ,x_p \}$. We prove that the invariant field $\mathbb{Q}(x_1,\ldots… Expand

ON NOETHER'S RATIONALITY PROBLEM FOR CYCLIC GROUPS OVER Q

- Mathematics
- 2016

Let p be a prime number. Let Cp, the cyclic group of order p, permute transitively a set of indeterminates {x1,...,xp}. We prove that the invariant field Q(x1,...,xp) Cp is rational over Q if and… Expand

On Noether's rationality problem for cyclic groups over Q

- Mathematics
- 2017

Let p be a prime number. Let C p , the cyclic group of order p , permute transitively a set of indeterminates f x 1 ;:::;x p g . We prove that the invariant eld Q ( x 1 ;:::;x p ) C p is rational… Expand

Heights and Principal Ideals of Certain Cyclotomic Fields

- Mathematics
- 2020

In this expository paper we present Plans’ 2016 proof of the fact that the primes l that split in \(\mathbf{Q}(\zeta _{l-1})\) into products of principal ideals, are… Expand

Noether's problem and rationality problem for multiplicative invariant fields: a survey.

- Mathematics
- 2020

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… Expand

Chow’s Theorem for Semi-abelian Varieties and Bounds for Splitting Fields of Algebraic Tori

- Mathematics
- Acta Mathematica Sinica, English Series
- 2019

A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper we generalize Chow's theorem to semi-abelian varieties. This contributes… Expand

#### References

SHOWING 1-10 OF 30 REFERENCES

Retract Rational Fields

- Mathematics
- 2009

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… Expand

Bogomolov multipliers and retract rationality for semi-direct products

- Mathematics
- 2012

Let $G$ be a finite group. The Bogomolov multiplier $B_0(G)$ is constructed as an obstruction to the rationality of $\bm{C}(V)^G$ where $G\to GL(V)$ is a faithful representation over $\bm{C}$. We… Expand

Reduction theorems for Noether's problem

- Mathematics
- 2007

Let K be any field, and G be a finite group. Let G act on the rational function field K(x(g) : g ∈ G) by K-automorphisms and h x(g) = x(hg). Denote by K(G) = K(x(g): g ∈ G) G the fixed field.… Expand

Frobenius groups and retract rationality

- Mathematics
- 2012

Abstract Let k be any field, G be a finite group acting on the rational function field k ( x g : g ∈ G ) by h ⋅ x g = x h g for any h , g ∈ G . Define k ( G ) = k ( x g : g ∈ G ) G . Noether’s… Expand

Cyclotomic fields with unique factorization.

- Mathematics
- 1976

2m For a natural number w>2, we let Cm = Q(e) be the ra-th cyclotornic field, of degree (m) over the field Q of rational numbers, we let hm denote the class number of Cm, and we let/? be a prime… Expand

ON PURELY-TRANSCENDENCY OF A CERTAIN FIELD

- Mathematics
- 1954

Let k be a field of characteristic p>G, and K=k(s,sn)a purely transcendental extension field over k. We denote by a the automorphis m of K induced by a cyclic permutation d:s>se1(i mod. p). The… Expand

Isoclinism and Stable Cohomology of Wreath Products

- Mathematics
- 2013

Using the notion of isoclinism introduced by P. Hall for finite p-groups, we show that many important classes of finite p-groups have stable cohomology detected by abelian subgroups (see Theorem 11).… Expand

On a problem of Chevalley

- Mathematics
- 1955

Recently Prof. Chevalley in Nagoya suggested to the author the following problem: Let k be a field, K 5 = k ( x 1 , x 2 , x 3 , x 4 , x 5 ) be a purely transcendental extension field (of… Expand

WEBER'S CLASS NUMBER PROBLEM IN THE CYCLOTOMIC ℤ2-EXTENSION OF ℚ, III

- Mathematics
- 2011

Let hn denote the class number of which is a cyclic extension of degree 2n over the rational number field ℚ. There are no known examples of hn > 1. We prove that a prime number l does not divide hn… Expand

Unramified Brauer groups of finite and infinite groups

- Mathematics
- 2012

The Bogomolov multiplier is a group theoretical invariant isomorphic to the unramified Brauer group of a given quotient space. We derive a homological version of the Bogomolov multiplier, prove a… Expand