Computing the Length of Sum of Squares and Pythagoras Element in a Global Field

TL;DR

This paper develops algorithms to compute the length of sums of squares and Pythagoras elements in global fields, advancing computational methods in algebraic number theory.

Contribution

It introduces new algorithms for calculating lengths in global fields and a procedure to construct Pythagoras elements, enhancing computational tools in the field.

Findings

Algorithms for length computation in non-dyadic completions

Algorithms for length computation in dyadic completions of number fields

Procedure for constructing Pythagoras elements with specified length

Abstract

This paper presents algorithms for computing the length of a sum of squares and a Pythagoras element in a global field $K$ of characteristic different from $2$. In the first part of the paper, we present algorithms for computing the length in a non-dyadic and dyadic (if $K$ is a number field) completion of $K$. These two algorithms serve as subsidiary steps for computing lengths in global fields. In the second part of the paper we present a procedure for constructing an element whose length equals the Pythagoras number of a global field, termed a Pythagoras element.