Аннотация:For all possible quadratic number fields K, we obtain a description of square-free polynomials f(x)∈K[x] of degree 4 such that √f has a periodic expansion into a continued fraction in the field of formal power series K((x)), while the elliptic field L=K(x)(√f) has a fundamental S-unit of degree m, 4≤m≤12, m≠11, where the set S consists of two conjugate valuations defined on the field L and related to the uniformizer x of the field K(x).