On real quadratic fields and periodic expansions (Q909706)

Let \(x=[a_ 1;a_ 2,a_ 3,...]\) be the regular continued fraction expansion of an irrational number \(x>1\) and put \(x_ j:=[a_{j+1};a_{j+2},a_{j+3},...]\), \(j\geq 0\). Then any real number z, \(0<z<1\) has a unique expansion of the form \[ z=\sum^{\infty}_{n=1}b_ n(x_ 0x_ 1...x_{n-1})^{-1}\quad when\quad 0\leq b_ n\leq a_ n. \] Now let x be a quadratic irrational number. Then z has a periodic expansion iff z belongs to the quadratic field \({\mathbb{Q}}(x)\). A similar result holds for expansions of the type \[ z=\sum^{\infty}_{n=1}c_ n(-1)^{n-1}(x_ 0x_ 1...x_{n- 1})^{-1}. \]