On the periodicity of an algorithm for p-adic continued fractions

In this paper we study the properties of an algorithm for generating continued fractions in the field of p-adic numbers $\mathbb{Q}_p$. First of all, we obtain an analogue of the Galois' Theorem for classical continued fractions. Then, we investigate the length of the preperiod for periodic expansions of square roots. Finally, we prove that there exist infinitely many square roots of integers in $\mathbb{Q}_p$ that have a periodic expansion with period of length four, solving an open problem left by Browkin.