Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields

Berarducci (2000) studied irreducible elements of the ring k((G<0))⊕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 truncation integer parts of any (non-archimedean) real closed field and generalize results of Berarducci and Pitteloud. To this end, we study the canonical integer part Neg (F ) ⊕ Z of any truncation closed subfield F of k((G)), where Neg (F ) := F ∩ k((G)), and work out in detail how the general case can be reduced to the case of archimedean G. In particular, we prove that k((G<0))⊕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. Finally, we apply our results to two important classes of exponential fields: exponential algebraic power series and exponential-logarithmic power series.