2017/10/19 by Travis Morrison, Morrison, Travis · 1 citation
Arts and Humanities · Computer Science · Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Logic (math.LO) #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1710.07357
openalex publication_date 2017/10/19 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
We show that for any square-free natural number n and any global field K\nwith (\char(K), n)=1 containing the nth roots of unity, the pairs\n(x,y)\∈ K^*\× K^* such that x is not a norm of K(\√[n]y)/K form\na diophantine set over K. We use the Hasse norm theorem, Kummer theory, and\nclass field theory to prove this result. We also prove that for any n\∈\n\ℕ and any global field K with (\char(K), n)=1,\nK^*\∖ K*n is diophantine over K. For a number field K, this is\na result of Colliot-Th 'el `ene and Van Geel, proved using results on the\nBrauer-Manin obstruction. Additionally, we prove a variation of our main\ntheorem for global fields K without the nth roots of unity, where we\nparametrize varieties arising from norm forms of cyclic extensions of K\nwithout any rational points by a diophantine set.\n