Mikhail Kotchetov : Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields

Posted by Matthias Aschenbrenner , part of the Logic Seminar.

At

Oct. 6, 2005, 4 p.m.

In

SEO 427

Abstract

Berarducci (2000) studied irreducible elements of the ring $k((G^{<0}))\oplus \Z$, which is an integer part of the power series field $k((G))$ where $G$ is an ordered divisible abelian group and $k$ is an ordered field. Pitteloud (2001) proved that some of the irreducible elements constructed by Berarducci are actually prime. Both authors mainly concentrated on the case of archimedean $G$. In this paper, we study {\it truncation integer parts} of any real closed field and generalize results of Berarducci and Pitteloud. In particular, we prove that $k((G^{<0}))\oplus\Z$ has (cofinally many) prime elements for any ordered divisible abelian group $G$. Addressing a question in the paper of Berarducci, we show that every truncation integer part of a non-archimedean exponential field has a cofinal set of irreducible elements. (Joint with S. Kuhlmann and D. Biljakovic.)

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