A factorisation theory for generalised power series and omnific integers

TL;DR

This paper develops a factorisation theory for generalized power series and omnific integers, establishing irreducibility results, bounds on factors, and solving a conjecture about primality within surreal numbers.

Contribution

It introduces a comprehensive factorisation framework for generalized power series and omnific integers, including bounds on irreducibles and solutions to existing conjectures.

Findings

Every series admits a finite irreducible factorisation with bounded number of factors.

The omnific integer ^{\u221a2} +  + 1 is proven to be prime.

New classes of irreducible and prime series and integers are identified.

Abstract

We prove that in every ring of generalised power series with non-positive real exponents and coefficients in a field of characteristic zero, every series admits a factorisation into finitely many irreducibles of infinite support, the number of which can be bounded in terms of the order type of the series, and a unique product, up to multiplication by a unit, of factors of finite support. We deduce analogous results for the ring of omnific integers within Conway's surreal numbers, using a suitable notion of infinite product. In turn, we solve Gonshor's conjecture that the omnific integer $\omega^{\sqrt{2}} + \omega + 1$ is prime. We also exhibit new classes of irreducible and prime generalised power series and omnific integers, generalising previous work of Berarducci and Pitteloud.