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


Author: Bernat Plans
Journal: Proc. Amer. Math. Soc. 145 (2017), 2407-2409
MSC (2010): Primary 12F10, 12F20, 13A50, 14E08, 11R18, 11R29
DOI: https://doi.org/10.1090/proc/13438
Published electronically: November 30, 2016
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Abstract: 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 ,x_p)^{C_p}$ is rational over $ \mathbb{Q}$ if and only if the $ (p-1)$-th cyclotomic field $ \mathbb{Q}(\zeta _{p-1})$ has class number one.


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Additional Information

Bernat Plans
Affiliation: Departament de Matemàtiques, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028 Barcelona, Spain
Email: bernat.plans@upc.edu

DOI: https://doi.org/10.1090/proc/13438
Keywords: Noether's problem, rational field extension, cyclic group, cyclotomic field
Received by editor(s): May 30, 2016
Received by editor(s) in revised form: August 3, 2016
Published electronically: November 30, 2016
Additional Notes: This research was partially supported by grant 2014 SGR-634 and grant MTM2015-66180-R
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2016 American Mathematical Society