= Solution
<Minkowski convex body theorem>[Minkowski's lemma] states that if $\Lambda$ is a full lattice in $\mathbb R^n$ and $C\subset\mathbb R^n$ is measurable, <convex set>[convex], and centrally symmetric, then
$$
\operatorname{vol}(C)>2^n\operatorname{covol}(\Lambda)
$$
implies that $C$ contains a nonzero point of $\Lambda$.
Let $\sigma_1,\sigma_2:K\to\mathbb R$ be the two real embeddings. The <Minkowski embedding of a real quadratic field> identifies $\mathcal O_K$ with a lattice
$$
\Lambda=\{(\sigma_1(\alpha),\sigma_2(\alpha)):\alpha\in\mathcal O_K\}
$$
of covolume $\sqrt{|D_K|}$. Choose a constant $c$ satisfying
$$
2\sqrt{|D_K|}<c<4\sqrt{|D_K|}.
$$
For $R>0$, consider the closed diamond
$$
C_R=
\left\{(x,y):\frac{|x|}{R}+\frac{|y|}{c/R}\leq1\right\}.
$$
It is convex and centrally symmetric, and its area is
$$
\operatorname{vol}(C_R)=2R\frac cR=2c>4\sqrt{|D_K|}.
$$
Minkowski's lemma supplies a nonzero $\alpha_R\in\mathcal O_K$ whose two embeddings lie in $C_R$.
For $(x,y)\in C_R$, the <arithmetic-geometric mean inequality> gives
$$
|xy|leq\frac14R\frac cR=\frac c4<\sqrt{|D_K|}.
$$
Since the coordinate product is the <field norm>,
$$
|N_{K/\mathbb Q}(\alpha_R)|<\sqrt{|D_K|}.
$$
Finally, $|\sigma_2(\alpha_R)|\leq c/R\to0$ as $R\to\infty$. No fixed nonzero <algebraic integer> can occur for arbitrarily large $R$, because its second embedding is nonzero. Consequently the elements $\alpha_R$ obtained along an unbounded sequence of $R$ contain infinitely many distinct values. Here $N(\alpha)$ denotes the absolute norm, as usual in this inequality.
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