A factorisation theory for generalised power series and omnific integers

We prove that in every ring of generalised power series with non-positive real exponents and coefficients in a field of char- acteristic zero, every series admits a factorisation into finitely many irreducibles with 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 with 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 theomnificintegerω 2+ω+1isprime. We also exhibit new classes of irreducible and prime gener- alised power series and omnific integers, generalising previous work of Berarducci and Pitteloud.