On the Quadratic Continued Fractions Over 𝔽 Q (X)

Abstract

Let 𝔽 q ((X −1)) be the field of formal power series in X −1 over 𝔽 q , the field with q elements. Let f ∈ 𝔽 q ((X −1)) satisfy the irreducible polynomial Af 2 + Bf + C = 0, with Δ =B 2 − 4AC. Let Per(f) be the length of the period of the continued fraction expansion of f. In this article, we show that We also prove that if Q is a monic polynomial with even degree, then the length of the period of the continued fraction expansion of any square root of Q is less than