Eisenstein integers are complex numbers of the form $z = a + b\omega$, where $a$ and $b$ are integers and $\omega = \frac{-1 + \sqrt{-3}}{2}$ is a primitive cube root of unity. This set of numbers forms a unique ring that has interesting properties, particularly in relation to factorization and ideal theory, connecting deeply with the structure of rings of integers in algebraic number fields and extending concepts found in Gaussian integers.
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