Final answer:
To show that ℤ[√(-2)] is a Euclidean domain, we need to prove that division with remainder is possible for any two elements in the set. By using the norm function N(x) = x^2 + 2y^2, where x and y are integers, we can show that the division with remainder is possible for any two elements in ℤ[√(-2)]. Thus, ℤ[√(-2)] is a Euclidean domain with the norm function N(x) = x^2 + 2y^2.
Step-by-step explanation:
To show that ℤ[√(-2)] is a Euclidean domain, we need to prove that division with remainder is possible for any two elements in the set. Let's assume we have two elements a and b in ℤ[√(-2)]. We want to find q and r such that a = bq + r, where r = 0 or N(r) < N(b). Here, N(x) represents the norm of the element x in ℤ[√(-2)].
By using the norm function N(x) = x2 + 2y2, where x and y are integers, we can show that the division with remainder is possible for any two elements in ℤ[√(-2)]. Thus, ℤ[√(-2)] is a Euclidean domain with the norm function N(x) = x2 + 2y2.