Real Algebraic and Analytic Geometry |

Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields.

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Submission: 2004, June 6.

*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 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) \oplus Z of
any truncation closed subfield F of k((G)), where Neg (F) := F \ 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)) \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 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.*

Mathematics Subject Classification (2000): 06F25, 13A16, 03H15, 03E10, 12J25, 13A05.

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