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- W4301321036 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 $nin mathbb{N}$ 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." @default.
- W4301321036 created "2022-10-05" @default.
- W4301321036 creator A5062486522 @default.
- W4301321036 date "2017-10-19" @default.
- W4301321036 modified "2023-10-16" @default.
- W4301321036 title "Diophantine definability of nonnorms of cyclic extensions of global fields" @default.
- W4301321036 doi "https://doi.org/10.48550/arxiv.1710.07357" @default.
- W4301321036 hasPublicationYear "2017" @default.
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