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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Diophantine definability of nonnorms of cyclic extensions of global fields

Author: Travis Morrison
Journal: Trans. Amer. Math. Soc. 372 (2019), 5825-5850
MSC (2010): Primary 11D57; Secondary 11U99
Published electronically: March 26, 2019
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Abstract: We show that, for any square-free natural number $ n$ and any global field $ K$ with $ ({\rm char}(K), n)=1$ containing a primitive $ n$th root of unity, the pairs $ (x,y)\in K^{\times }\times K^{\times }$ such that $ x$ is not a relative norm of $ K(\sqrt [n]{y})/K$ form a diophantine set over $ K$. We use the Hasse norm theorem, Kummer theory, and class field theory to prove this result. We also prove that, for any $ n\in {\mathbb{N}}$ and any global field $ K$ with $ {\rm char}(K)\neq n$, $ K^{\times }\setminus K^{\times n}$ is diophantine over $ K$. For a number field $ K$, this is a result of Colliot-Thélène and Van Geel, proved using results on the Brauer-Manin obstruction. Additionally, we prove a variation of our main theorem for global fields $ K$ without the $ n$th roots of unity, where we parametrize varieties arising from norm forms of cyclic extensions of $ K$ without any rational points by a diophantine set.

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Additional Information

Travis Morrison
Affiliation: Institute for Quantum Computing, The University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada

Received by editor(s): October 19, 2017
Received by editor(s) in revised form: November 1, 2018, and November 9, 2018
Published electronically: March 26, 2019
Additional Notes: The author was partially supported by National Science Foundation grants DMS-1056703 and CNS-1617802, and in part by funding from the Natural Sciences and Engineering Research Council of Canada, the Canada First Research Excellence Fund, CryptoWorks21, Public Works and Government Services Canada, and the Royal Bank of Canada.
Article copyright: © Copyright 2019 American Mathematical Society