Submission date: 19 October 2017

Abstract:

We show that for any square-free natural number n and any global field K with (char(K), n)=1 containing the nth roots of unity, the pairs (x,y) in K^* × K^* such that x is not a 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 N and any global field K with char(K) ≠ n, K^* \ K^{*n} is diophantine over K. For a number field K, this is a result of Colliot-Théléne 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 nth 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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Full text arXiv 1710.07357: pdf, ps.