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Truncation integer parts of valued fields

Presented by: 
S Kuhlmann [Saskatchewan]
Date: 
Wednesday 11th May 2005 - 11:30 to 12:30
Venue: 
INI Seminar Room 2
Abstract: 

We study the problem of constructing complements to the valuation ring of a valued field, which are $k$-algebras over the residue field $k$.

These $k$-algebras have many important arithmetic properties, and are closely related to truncation integer parts.

We work out the case of henselian fields and study the canonical integer part $\Neg(F)\oplus\Z$ of any truncation closed subfield $F$ of the field of power series $k((G))$, where $\Neg(F):=F\cap k((G^{<0}))$. 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 a 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.

University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons