English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $G$ be a finite group acting on the rational function field$\mathbb{C}(x_g : g\in G)$ by $\mathbb{C}$-automorphisms $h(x_g)=x_{hg}$ forany $g,h\in G$. Noether's problem asks whether the invariant field$\mathbb{C}(G)=k(x_g : g\in G)^G$ is rational (i.e. purely transcendental) over$\mathbb{C}$. Saltman and Bogomolov, respectively, showed that for any prime$p$ there exist groups $G$ of order $p^9$ and of order $p^6$ such that$\mathbb{C}(G)$ is not rational over $\mathbb{C}$ by showing the non-vanishingof the unramified Brauer group: $Br_{nr}(\mathbb{C}(G))\neq 0$. For $p=2$, Chu,Hu, Kang and Prokhorov proved that if $G$ is a 2-group of order $\leq 32$, then$\mathbb{C}(G)$ is rational over $\mathbb{C}$. Chu, Hu, Kang and Kunyavskiishowed that if $G$ is of order 64, then $\mathbb{C}(G)$ is rational over$\mathbb{C}$ except for the groups $G$ belonging to the two isoclinism families$\Phi_{13}$ and $\Phi_{16}$. Bogomolov and B\"ohning's theorem claims that if$G_1$ and $G_2$ belong to the same isoclinism family, then $\mathbb{C}(G_1)$and $\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic. We investigate thebirational classification of $\mathbb{C}(G)$ for groups $G$ of order 128 with$Br_{nr}(\mathbb{C}(G))\neq 0$. Moravec showed that there exist exactly 220groups $G$ of order 128 with $Br_{nr}(\mathbb{C}(G))\neq 0$ forming 11isoclinism families $\Phi_j$. We show that if $G_1$ and $G_2$ belong to$\Phi_{16}, \Phi_{31}, \Phi_{37}, \Phi_{39}, \Phi_{43}, \Phi_{58}, \Phi_{60}$or $\Phi_{80}$ (resp. $\Phi_{106}$ or $\Phi_{114}$), then $\mathbb{C}(G_1)$ and$\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic with$Br_{nr}(\mathbb{C}(G_i))\simeq C_2$. Explicit structures of non-rationalfields $\mathbb{C}(G)$ are given for each cases including also the case$\Phi_{30}$ with $Br_{nr}(\mathbb{C}(G))\simeq C_2\times C_2$.

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