A p-adic analogue of Siegel's Theorem on sums of squares

Dittmann, Philip and Anscombe, Sylvy orcid iconORCID: 0000-0002-9930-2804 (2020) A p-adic analogue of Siegel's Theorem on sums of squares. Mathematische Nachrichten, 293 (8). ISSN 0025-584X

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Official URL: https://doi.org/10.1002/mana.201900173

Abstract

Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p ‐adic Kochen operator provides a p ‐adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p ‐integral elements of K . We use this to formulate and prove a p ‐adic analogue of Siegel's theorem, by introducing the p ‐Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p ‐Pythagoras number and show that the growth of the p ‐Pythagoras number in finite extensions is bounded.


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