On the finiteness of $\mathfrak{P}$-adic continued fractions for number fields
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866917108582973440 |
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| author | Capuano, Laura Murru, Nadir Terracini, Lea |
| author_facet | Capuano, Laura Murru, Nadir Terracini, Lea |
| contents | 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 $α\in K$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2105_12570 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the finiteness of $\mathfrak{P}$-adic continued fractions for number fields Capuano, Laura Murru, Nadir Terracini, Lea Number Theory 11J70, 11D88, 11Y65, 11J72 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 $α\in K$ 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. |
| title | On the finiteness of $\mathfrak{P}$-adic continued fractions for number fields |
| topic | Number Theory 11J70, 11D88, 11Y65, 11J72 |
| url | https://arxiv.org/abs/2105.12570 |