Continued fractions for quadratic elements in formal power series

Abstract

There exist certain quadratic elements α∈ℚ((t ⁻¹)) over the rational function field ℚ(t) having nonperiodic continued fraction expansion, see W.M. Schmidt in (Acta Arith. 95(2):139–166, 2000). Hence we need a modification of Lagrange’s theorem with regard to function fields instead of number fields. In this paper, we introduce a class of continued fractions and describe Lagrange’s theorem as a conjecture related to quadratic elements over ℚ(t). We give some examples which support our conjecture.