Extended gcd of quadratic integers

where θ = 1 + √ d 2 if d mod 4 = 1 and θ = √ d if d mod 4 = 2, 3. The purpose of this paper is to compute the extended gcd of to quadratic integers in ring Z[θ]. We assume throughout that Z[θ] is principal ideal ring, but not necessarily an euclidean ring. If [a, b+ cθ] is the module {ax+(b+ cθ)y, x, y ∈ Z}, it can be shown [3] that I is an ideal of Z[θ] if and only if I = [a, b+ cθ]; where a, b, c ∈ Z with c|b, c|a and ac|N(b+ cθ), (N(b+ cθ) the norm of b+ cθ). It is known [?, Samuel]hat any non-zero ideal of Z[θ] is a free Z-module of range 2 If I = Zα + Z(β + γθ) is an ideal of Z[θ], then I have a generator for which the absolute value of the norm is equal to the norm N(I) of I (where N(I) = |α× γ|). Conversely any X in I such that |N(X)| = N(I) is a generator of I. 2-Algorithm of the extended gcd Let I = [a + bθ], J = [c + dθ] non-zero ideals of Z[θ], then I + J = [m + nθ] is an ideal, and m+nθ = gcd((a+ bθ), (c+ dθ)). This gcd is defined modulo a unity of Q( √ d). There exist U, V ∈ Z[θ] such that