Anscombe, Sylvy ORCID: 0000-0002-9930-2804, Dittmann, Philip and Fehm, Arno
(2019)
A p-adic analogue of Siegel’s theorem on sums of squares.
Mathematische Nachrichten
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ISSN 0025-584X
(Submitted)
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Official URL: https://onlinelibrary.wiley.com/journal/15222616
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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