= Solution
The <Eisenstein integers> are $\mathbb Z[\omega]$, where $\omega^2+\omega+1=0$. Their multiplicative <Eisenstein-integer norm>[algebraic norm] is
$$
N(a+b\omega)=(a+b\omega)(a+b\overline\omega)=a^2-ab+b^2.
$$
Given $\alpha,\beta\ne0$, choose an Eisenstein integer $q$ nearest to the complex number $\alpha/\beta$. The hexagonal lattice has covering radius $1/\sqrt3$, so
$$
\left|\frac\alpha\beta-q\right|\leq\frac1{\sqrt3}<1.
$$
With $r=\alpha-q\beta$, multiplicativity gives
$$
N(r)=N(\beta)\left|\frac\alpha\beta-q\right|^2<N(\beta).
$$
Thus division with remainder decreases the nonnegative integer norm, proving that $\boxed{\mathbb Z[\omega]\text{ is a Euclidean domain}}$.
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