A p‐adic analogue of Siegel's theorem on sums of squares

Sylvy Anscombe, Philip Dittmann and Arno Fehm

Mathematische Nachrichten, 2020, vol. 293, issue 8, 1434-1451

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.

Date: 2020
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https://doi.org/10.1002/mana.201900173

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