= Solution
Embed the <ring of integers> into $\mathbb R^2$ by its two real embeddings. Its image is a full <Euclidean lattice> of some covolume $d>0$. Fix $C>d$ and consider the symmetric rectangle $|\sigma_1(\alpha)|\le T$, $|\sigma_2(\alpha)|\le C/T$. Its area is $4C>4d$, so the <Minkowski convex body theorem> gives a nonzero <algebraic integer> $\alpha_T$ in it. Its <algebraic norm> is a nonzero <integer> with $|N(\alpha_T)|\le C$.
As $T\to\infty$, infinitely many distinct $\alpha_T$ must occur: a finite collection of nonzero elements has a positive minimum of $|\sigma_2(\alpha)|$, whereas here that coordinate tends to zero. The <principal ideals> $(\alpha_T)$ have norm $|N(\alpha_T)|\le C$. There are only finitely many ideals of bounded norm: they are <subgroups> of bounded index in a rank-two <free abelian group>, and <integer> <Hermite normal forms> give finitely many possibilities at each such index. Thus infinitely many of these elements generate one ideal. Ratios of two generators of the same ideal are units, because inclusion in both directions supplies inverse <algebraic integers>. Among infinitely many distinct generators choose two whose ratio is not $\pm1$. \b[This gives a non-trivial unit without assuming Dirichlet's theorem.]
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