Definition. An algebraic number field $K$ is a norm-Euclidean number field, if for every pair $(\alpha,\,\beta)$ of the integers of $K$ , where $\beta \neq 0$ , there exist integers $\varkappa$ and $\varrho$ of the field such that $$\alpha = \varkappa\beta+\varrho,\;\; |\mbox{N}(\varrho)| < |\mbox{N}(\beta)|.$$ Here $\mbox{N}$ means the norm function in $K$ .

Theorem 1. A field $K$ is norm-Euclidean if and only if each number $\gamma$ of $K$ is expressible in the form

(1)

Proof. First assume the condition (1). Let $\alpha$ and $\beta$ be integers of $K$ , $\beta \neq 0$ . Then there are the numbers $\varkappa,\,\delta \in K$ such that $\varkappa$ is integer and $$\frac{\alpha}{\beta} = \varkappa+\delta, \quad |\mbox{N}(\delta)| < 1.$$ Thus we have $$\alpha = \varkappa\beta+\beta\delta = \varkappa\beta+\varrho.$$ Here $\varrho = \beta\delta$ is integer, since $\alpha$ and $\varkappa\beta$ are integers. We also have $$|\mbox{N}(\varrho)| = |\mbox{N}(\beta)|\cdot|\mbox{N}(\delta)| < |\mbox{N}(\beta)|\cdot1 = |\mbox{N}(\beta)|.$$ Accordingly, $K$ is a norm-Euclidean number field. Secondly assume that $K$ is norm-Euclidean. Let $\gamma$ be an arbitrary element of the field. We can determine a rational integer $m\,(\neq 0)$ such that $m\gamma$ is an algebraic integer of $K$ . The assumption guarantees the integers $\varkappa$ , $\varrho$ of $K$ such that $$m\gamma = \varkappa m+\varrho, \quad \mbox{N}(\varrho) < \mbox{N}(m).$$ Thus $$\gamma = \frac{m\gamma}{m} = \varkappa+\frac{\varrho}{m}, \quad \left\vert\mbox{N} \left(\frac{\varrho}{m}\right)\right\vert = \frac{|\mbox{N}(\varrho)|}{|\mbox{N}(m)|} < 1,$$ Q.E.D.

Theorem 2. In a norm-Euclidean number field, any two non-zero integers have a greatest common divisor.

Proof. We recall that the greatest common divisor of two elements of a commutative ring means such a common divisor of the elements that it is divisible by each common divisor of the elements. Let now $\varrho_0$ and $\varrho_1$ be two algebraic integers of a norm-Euclidean number field $K$ . According the definition there are the integers $\varkappa_i$ and $\varrho_i$ of $K$ such that

Remark. The ring of integers of any norm-Euclidean number field is a unique factorization domain and thus all ideals of the ring are principal ideals. But not all algebraic number fields with ring of integers a UFD are norm-Euclidean, e.g. $\mathbb{Q}(\sqrt{14})$ .

Theorem 3. The only norm-Euclidean quadratic fields $\mathbb{Q}(\sqrt{d})$ are those with $$d\in\{-11,\,-7,\,-3,\,-2,\,-1,\,2,\,3,\,5,\,6,\,7,\,11,\,13,\,17,\,19,\,21,\,29,\,33,\,37,\,41,\,57,\,73\}.$$