Quadratic irrationality

A root of a quadratic trinomial with rational coefficients which is irreducible over the field of rational numbers. A quadratic irrationality is representable in the form $a+b\sqrt{d}$, where $a$ and $b$ are rational numbers, $b\ne 0$, and $d$ is an integer which is not a perfect square. A real number $\alpha$ is a quadratic irrationality if and only if it has an infinite periodic continued fraction expansion.

References

[a1]	A.Ya. Khinchin, "Continued fractions" , Phoenix Sci. Press (1964) pp. Chapt. II, §10 (Translated from Russian)