Eisenstein Integer

The Eisenstein integers, sometimes also called the Eisenstein-Jacobi integers (Finch 2003, p. 601), are numbers of the form , where and are normal integers,

(1)

is one of the roots of , the others being 1 and

(2)

The sums, differences, and products of Eisenstein integers is another Eisenstein integer.

Eisenstein integers are complex numbers that are members of the imaginary quadratic field , which is precisely the ring (Wagon 1991, p. 320). The field of Eisenstein integers has the six units (or roots of unity), namely , , and (Wagon 1991, p. 320; Guy 1994, p. 35).

Every nonzero Eisenstein integer has a unique (up to ordering) factorization up to associates, where associates are Eisenstein integers related to the given Eisenstein integer by rotations of multiples of in the complex plane. Specifically, any nonzero Eisenstein integer is uniquely the product of powers of , , and the "positive" Eisenstein primes, where the "positive" Eisenstein integers are those falling within the triangular wedge illustrated above (Conway and Guy 1996).

The analog of Fermat's theorem for Eisenstein integers is that a prime number can be written in the form

(3)

iff . These are precisely the primes of the form (Conway and Guy 1996).

Every Eisenstein integer is within a distance of some multiple of a given Eisenstein integer .

Dörrie (1965) uses the alternative notation

			(4)
			(5)

for and , and calls numbers of the form -numbers. and satisfy

			(6)
			(7)
			(8)
			(9)
			(10)
			(11)