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

## TL;DR

This paper introduces a p-adic analogue of Siegel's theorem, demonstrating that the p-Pythagoras number is uniformly bounded across number fields using the p-adic Kochen operator.

## Contribution

It formulates and proves a p-adic version of Siegel's theorem, establishing bounds on the p-Pythagoras number across number fields.

## Key findings

- p-Pythagoras number is uniformly bounded across number fields
- Growth of p-Pythagoras number in finite extensions is bounded
- Introduces a p-adic analogue of Siegel's theorem

## 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.