Multiplicativity of the norm $N_{K/\QQ}$ and units of $\OO_K$

Let \(K\) be a number field with ring of integers \(\OO_K\). For \(a \in K\), let \(N(a) \coloneqq \det(m_a)\), where \(m_a\colon K \to K\) is the \(\QQ\)-linear map \(x \mapsto ax\); this defines the norm \(N = N_{K/\QQ}\colon K \to \QQ\).

• For all \(a, b \in K\), \(N(ab) = N(a)N(b)\).

• \(N(\OO_K) \subseteq \ZZ\).

• For \(a, b \in \OO_K\), if \(a\) divides \(b\) in \(\OO_K\), then \(N(a) \divides N(b)\) in \(\ZZ\).

• For \(a \in \OO_K\), \(a\) is a unit of \(\OO_K\) if and only if \(N(a) = \pm 1\).