## Asian Journal of Mathematics

- Asian J. Math.
- Volume 17, Number 4 (2013), 689-714.

### Noether's problem and unramified Brauer groups

Akinari Hoshi, Ming-Chang Kang, and Boris E. Kunyavskii

#### Abstract

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 transcendental) over $k$. It is known that, if $\mathbb{C}(G)$ is rational over $\mathbb{C}$, then $B_0(G) = 0$ where $B_0(G)$ is the unramified Brauer group of $\mathbb{C}(G)$ over $\mathbb{C}$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B_0(G) = 0$. This result was disproved by Moravec for $p = 3, 5, 7$ by computer calculations. We will prove the following theorem. Theorem. Let $p$ be any odd prime number, $G$ be a group of order $p^5$. Then $B_0(G) \neq 0$ if and only if $G$ belongs to the isoclinism family ${\Phi}_{10}$ in R. James's classification of groups of order $p^5$.

#### Article information

**Source**

Asian J. Math., Volume 17, Number 4 (2013), 689-714.

**Dates**

First available in Project Euclid: 22 August 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ajm/1408712350

**Mathematical Reviews number (MathSciNet)**

MR3152260

**Zentralblatt MATH identifier**

1291.13012

**Subjects**

Primary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24] 14E08: Rationality questions [See also 14M20] 14M20: Rational and unirational varieties [See also 14E08] 20J06: Cohomology of groups 12F12: Inverse Galois theory

**Keywords**

Noether’s problem rationality problem unramified Brauer groups Bogomolov multipliers rationality retract rationality

#### Citation

Hoshi, Akinari; Kang, Ming-Chang; Kunyavskii, Boris E. Noether's problem and unramified Brauer groups. Asian J. Math. 17 (2013), no. 4, 689--714. https://projecteuclid.org/euclid.ajm/1408712350