Diophantine definability of nonnorms of cyclic extensions of global fields

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 $(x,y)\in K^{\ast }\times K^{\ast }$ such that $x$ is not a norm of $K(\sqrt[n]{y})\mathrm{/}K$ 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}$$andanyglobalfield$K$with$($\text{char}$(K), n)=1$,$K^*\setminus K^{*n}$isdiophantineover$K$\mathrm{.}Foranumberfield$K$, 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.