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引用次数: 0
摘要
对于数域$K$的整数环的素数理想$\mathfrak{P}$,给出$\mathfrak{P}$-进连续分数的一般定义,其中还包括$ P $-进数域内连分数的经典定义。我们给出了K$的一些充要条件,以保证K$中的每个$\alpha\对于除有限个$\mathfrak{P}$以外的所有$\mathfrak{P}$有一个有限的$\mathfrak{P}$进进的连分式展开,解决了Rosen在阿基米德环境中提出的类似问题。
On the finiteness of $\mathfrak{P}$-adic continued fractions for number fields
For a prime ideal $\mathfrak{P}$ of the ring of integers of a number field $K$, we give a general definition of $\mathfrak{P}$-adic continued fraction, which also includes classical definitions of continued fractions in the field of $p$--adic numbers. We give some necessary and sufficient conditions on $K$ ensuring that every $\alpha\in K$ admits a finite $\mathfrak{P}$-adic continued fraction expansion for all but finitely many $\mathfrak{P}$, addressing a similar problem posed by Rosen in the archimedean setting.
期刊介绍:
The Bulletin de la Société Mathématique de France was founded in 1873, and it has published works by some of the most prestigious mathematicians, including for example H. Poincaré, E. Borel, E. Cartan, A. Grothendieck and J. Leray. It continues to be a journal of the highest mathematical quality, using a rigorous refereeing process, as well as a discerning selection procedure. Its editorial board members have diverse specializations in mathematics, ensuring that articles in all areas of mathematics are considered. Promising work by young authors is encouraged.