Diophantine definability of nonnorms of cyclic extensions of global fields

Abstract:

We show that, for any square-free natural number $n$ and any global field $K$ with $(\textrm {char}(K), n)=1$ 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 $\textrm {char}(K)\neq n$, $K^{\times }\setminus K^{\times n}$ 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.