On the problem of periodicity of continued fractions in hyperelliptic fields

We present new results concerning the problem of periodicity of continued fractions which are expansions of quadratic irrationalities in a field , where is a field of characteristic different from 2, , . Let be a square-free polynomial and suppose that the valuation of the field has two extensions and to the field . We set . A deep connection between the periodicity of continued fractions in the field and the existence of -units made it possible to make great advances in the study of periodic and quasiperiodic elements of the field , and also in problems connected with searching for fundamental -units. Using a new efficient algorithm to search for solutions of the norm equation in the field we manage to find examples of periodic continued fractions of elements of the form , which is a fairly rare phenomenon. For the case of an elliptic field , , we describe all square-free polynomials with a periodic expansion of into a continued fraction in the field . Bibliography: 16 titles.