ON THE FINITENESS OF p-ADIC CONTINUED FRACTIONS FOR NUMBER FIELDS

For a prime ideal p of the ring of integers of a number field K, we give a general definition of a 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 for all but finitely many p, every alpha is an element of K admits a finite p-adic continued fraction expansion, addressing a similar problem posed by Rosen in the archimedean setting.