Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields.
Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields.
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实闭域的截断整数部分中的素数和不可约数。
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
S. Kuhlmann
中科院分区：
文献类型：
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作者：
D. Biljaković;M. Kochetov;S. Kuhlmann
Berarducci(2000)研究了环k((G<；0))‘Z的不可约元素，它是幂级数域k((G))的整数部分，其中G是有序可除交换群，k是有序域。Pitteloud(2001)证明了Berarducci构造的一些不可约元素实际上是素数。本文研究了任意(非阿基米德)实闭域的截断整数部分，推广了Berarducci和Pitteloud的结果。为此，我们研究了k((G))的任一截断闭子域F的标准整数部分Neg(F)‘Z，其中Neg(F)：=F\k((G<；0))，并详细地计算出如何将一般情况化为阿基米德G的情形。0)‘Z对任何有序可分交换群G都有(相当多)素元。在回答Berarducci文中的一个问题时，我们证明了非阿基米德指数域的每个截断整数部分都有一个不可约元素的余集。最后，我们将我们的结果应用于两类重要的指数域：指数代数幂级数和指数-对数幂级数。
Berarducci (2000) studied irreducible elements of the ring k((G <0 ))'Z, which is an integer part of the power series fleld k((G)) where G is an ordered divisible abelian group and k is an ordered fleld. 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 fleld and generalize results of Berarducci and Pitteloud. To this end, we study the canonical integer part Neg(F) 'Z of any truncation closed subfleld 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 )) 'Z has (coflnally 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 fleld has a coflnal set of irreducible elements. Finally, we apply our results to two important classes of exponential flelds: exponential algebraic power series and exponential-logarithmic power series.