Units and periodic Jacobi-Perron algorithms in real algebraic number fields of degree $3$

Abstract:

It is not known whether or not the Jacobi-Perron Algorithm of a vector in ${R_{n - 1}},n \geqslant 3$, whose components are algebraic irrationals, always becomes periodic. The author enumerates, from his previous papers, a few infinite classes of real algebraic number fields of any degree for which this is the case. Periodic Jacobi-Perron Algorithms are important, because they can be applied, inter alia, to calculate units in the corresponding algebraic number fields. The main result of this paper is expressed in the following theorem: There are infinitely many real cubic fields $Q(w),{w^3}$ cubefree, a and T natural numbers, such that the Jacobi-Perron Algorithm of the vector $(w,{w^2})$ becomes periodic; the length of the primitive preperiod is four, the length of the primitive period is three; a fundamental unit of $Q(w)$ is given by $e = {a^3}T + 1 - aw$.