Sums of integral squares in complex bi-quadratic fields and in CM fields

TL;DR

This paper investigates sums of squares in complex bi-quadratic and CM fields, proving that every algebraic integer in certain cases can be expressed as a sum of a limited number of squares, and establishing bounds on the Pythagoras number.

Contribution

It establishes new results on sums of squares in complex bi-quadratic and CM fields, including bounds on the Pythagoras number and specific classes where the number is exactly three.

Findings

Every algebraic integer in certain bi-quadratic fields is a sum of squares.

Every element of 4 times the ring of integers can be written as a sum of five squares.

The Pythagoras number of CM fields' rings of integers is at most five.

Abstract

Let $K$ be a complex bi-quadratic field with ring of integers $\mathcal{O}_{K}$. For $K = \mathbb{Q}(\sqrt{-m}$, $\sqrt{n}$), where $ m \equiv 3 \pmod 4 $ and $ n \equiv 1 \pmod 4$, we prove that every algebraic integer can be written as sum of integral squares. Using this, we prove that for any complex bi-quadratic field $K$, every element of $4\mathcal{O}_K$ can be written as sum of five integral squares. In addition, we show that the Pythagoras number of ring of integers of any CM field is at most five. Moreover, we give two classes of complex bi-quadratic fields for which $p(\mathcal{O}_{K})= 3$ and $p(4\mathcal{O}_{K})=3 $ respectively. Here, $p(\mathcal{O}_{K})$ is the Pythagoras number of ring of integers of $K$ and $p(4\mathcal{O}_{K})$ is the smallest positive integer $t$ such that every element of $4\mathcal{O}_{K}$ can be written as sum of $t$ integral squares.