Exercise 7, § 1

Given an integral domain A, we call a map w from $A' = A - \{0\}$ into N a Euclidean function if it satisfies the following conditions : $(S_I)$ w(xy) ≥ w(y) for each pair x, y of elements of A'. $(S_{II})$ If a ∈ A' and b ∈ A', then there exist elements q, r of A such that a = bq + r, and such that either $r = 0$ or $w(r) < w(b)$.

If there exists a Euclidean function on A then A is called a Euclidean domain. a) Show that Z and K[X ] (where K is a field) are Euclidean domains. b) Show that every Euclidean domain is a principal ideal domain (if a is an ideal of A, choose $a \in a$ such that $w(a)$ is as small as possible). c) Show that the ring A of Gaussian integers (VII, p. 1) is a Euclidean domain with respect to the function $w(a + bi) = a^2 + b^2 = N_{\mathbf{Q}(i)/\mathbf{Q}}(a + bi)$ (notice that, for all $x \in \mathbf{Q}(i)$, there exists $y \in A$ such that $N(x - y) \leq 112$). d) Show that the quadratic extension ring B of Z with basis $(1, e)$ ($e' \equiv 2$), is a Euclidean domain with respect to the function

(for all $x \in \mathbf{Q}(e)$, there exists $y \in B$ such that $|N(x - y)| \leq 112$)