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 $(\text{char}(K), n)=1$ containing the $n$th roots of unity, the pairs $(x,y)\in K^*\times K^*$ 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 $n\in \mathbb{N}$ and any global field $K$ with $(\text{char}(K), n)=1$, $K^*\setminus K^{*n}$ is diophantine over $K$. For a number field $K$, this is a result of Colliot-Th\'el\`ene 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.