Diophantine definability of nonnorms of cyclic extensions of global fields

TL;DR

This paper proves that the set of elements in global fields that are not norms from cyclic extensions is diophantine, extending previous results and using advanced number theory tools like the Hasse norm theorem and class field theory.

Contribution

It establishes the diophantine definability of nonnorms in cyclic extensions over global fields, including cases without the roots of unity, broadening prior work.

Findings

Non-norm elements form a diophantine set over global fields.

The set of non-nth powers is diophantine over global fields.

Extended results to fields lacking n-th roots of unity.

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.