engleski

Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields

Berarducci (2000) studied irreducible elements of the ring k((G<0))&#8853; 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 con- centrated 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) &#8853; Z of any truncation closed subfield F of k((G)), where Neg (F) := F &#8745; k((G<0)), 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)) &#8853; 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 expo- nential 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.

Prime; irreducible; truncation; integer part; real closed field; generalized power series; exponential integer part

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