Submission date: 6 April 2019

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.

Mathematics Subject Classification: 11E25, 12D15, 11S99, 11U09

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Full text arXiv 1904.03466: pdf, ps.