Pythagoras numbers for infinite algebraic fields
Number Theory
2026-02-27 v2
Abstract
We prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally real algebraic fields whose rings of integers have finite Pythagoras numbers, namely, one, two, three, and at least four.
Cite
@article{arxiv.2502.11222,
title = {Pythagoras numbers for infinite algebraic fields},
author = {Nicolas Daans and Stevan Gajović and Siu Hang Man and Pavlo Yatsyna},
journal= {arXiv preprint arXiv:2502.11222},
year = {2026}
}
Comments
11 pages; comments are welcome!