Factorising generalised power series and omnific integers

Vincenzo Mantova (Leeds)

Frank Adams 1,

While constructing surreal numbers, Conway introduced the subring of "omnific integers", and conjectured the existence of irreducible numbers, as well as a suitable kind of uniqueness of the factorisation. Berarducci eventually answered affirmatively to the first conjecture by creating an ordinal-valued (semi)valuation on rings of generalised power series.

In a joint work with S. L'Innocente, we tack on an extra step to produce a finer ordinal-valued valuation called "degree", and we prove that every generalised power series factors into some irreducibles and a unique factor with finite support. The factorisation is obtained by playing with the RV monoid, which is a quotient naturally appearing in the theory of valued fields.

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