A factorisation theory for generalised power series and omnific integers
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2017
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| _version_ | 1866911788085280768 |
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| author | L'Innocente, Sonia Mantova, Vincenzo |
| author_facet | L'Innocente, Sonia Mantova, Vincenzo |
| contents | 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 $ω^{\sqrt{2}} + ω+ 1$ is prime.
We also exhibit new classes of irreducible and prime generalised power series and omnific integers, generalising previous work of Berarducci and Pitteloud. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1710_07304 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | A factorisation theory for generalised power series and omnific integers L'Innocente, Sonia Mantova, Vincenzo Logic Rings and Algebras 13F25, 13F15 (primary), 13A05, 03E10 (secondary) 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 $ω^{\sqrt{2}} + ω+ 1$ is prime. We also exhibit new classes of irreducible and prime generalised power series and omnific integers, generalising previous work of Berarducci and Pitteloud. |
| title | A factorisation theory for generalised power series and omnific integers |
| topic | Logic Rings and Algebras 13F25, 13F15 (primary), 13A05, 03E10 (secondary) |
| url | https://arxiv.org/abs/1710.07304 |