Eisenstein Integer

The numbers $a+b\omega$, where

$$\omega\equiv {\textstyle{1\over 2}}(-1+i\sqrt{3}\,)$$

is one of the Roots of $z^3=1$, the others being 1 and

$$\omega^2={\textstyle{1\over 2}}(-1-i\sqrt{3}\,).$$

Eisenstein integers are members of the Quadratic Field $\Bbb{Q}(\sqrt{-3}\,)$, and the Complex Numbers ${\Bbb{Z}}[\omega]$. Every Eisenstein integer has a unique factorization. Specifically, any Nonzero Eisenstein integer is uniquely the product of Powers of $-1$, $\omega$, and the ``positive'' Eisenstein Primes (Conway and Guy 1996). Every Eisenstein integer is within a distance $\vert n\vert/\sqrt{3}$ of some multiple of a given Eisenstein integer $n$.

Dörrie (1965) uses the alternative notation

$$J \equiv {\textstyle{1\over 2}}(1+i\sqrt{3}\,)$$ (1)

$$O \equiv {\textstyle{1\over 2}}(1-i\sqrt{3}\,).$$ (2)

for $-\omega^2$ and $-\omega$, and calls numbers of the form $aJ+bO$ G-Number. $O$ and $J$ satisfy

$$J+O = 1$$ (3)

$$JO = 1$$ (4)

$$J^2+O = 0$$ (5)

$$O^2+J = 0$$ (6)

$$J^3 = -1$$ (7)

$$O^3 = -1.$$ (8)

The sum, difference, and products of $G$ numbers are also $G$ numbers. The norm of a $G$ number is

$$N(aJ+bO)=a^2+b^2-ab.$$ (9)

The analog of Fermat's Theorem for Eisenstein integers is that a Prime Number $p$ can be written in the form

$$a^2-ab+b^2=(a+b\omega)(a+b\omega^2)$$

Iff $3\notdiv p+1$. These are precisely the Primes of the form $3m^2+n^2$ (Conway and Guy 1996).

See also Eisenstein Prime, Eisenstein Unit, Gaussian Integer, Integer

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.

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

Dörrie, H. ``The Fermat-Gauss Impossibility Theorem.'' §21 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 96-104, 1965.

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.

Riesel, H. Appendix 4 in Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, 1994.

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