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Eisenstein Prime


Let $\omega$ be the Cube Root of unity $(-1+i\sqrt{3}\,)/2$. Then the Eisenstein primes are

1. Ordinary Primes Congruent to 2 (mod 3),

2. $1-\omega$ is prime in ${\Bbb{Z}}[\omega]$,

3. Any ordinary Prime Congruent to 1 (mod 3) factors as $\alpha\alpha^*$, where each of $\alpha$ and $\alpha^*$ are primes in ${\Bbb{Z}}[\omega]$ and $\alpha$ and $\alpha^*$ are not ``associates'' of each other (where associates are equivalent modulo multiplication by an Eisenstein Unit).


Cox, D. A. §4A in Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1989.

Guy, R. K. ``Gaussian Primes. Eisenstein-Jacobi Primes.'' §A16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 33-36, 1994.

Wagon, S. ``Eisenstein Primes.'' Mathematica in Action. New York: W. H. Freeman, pp. 278-279, 1991.

© 1996-9 Eric W. Weisstein