Eisenstein integer
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form where and are integers and is a primitive (hence non-real) cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane. The Eisenstein integers are a countably infinite set. Properties. The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field formula_3 — the third cyclotomic field. To see that  the Eisenstein integers are algebraic integers note that each is a root of the monic polynomial In particular, satisfies the equation The product of two Eisenstein integers and is given explicitly by The 2-norm of an Eisenstein integer is just its squared modulus, and is given by which is clearly a positive ordinary (rational) integer. Also, the complex conjugate of satisfies The group of units in this ring is the cyclic group formed by the sixth roots of unity in the complex plane: formula_9 the Eisenstein integers of norm 1. Eisenstein primes. If and are Eisenstein integers, we say  that divides if there is some Eisenstein integer such that . A non-unit Eisenstein integer is said to be an Eisenstein prime if its only non-unit divisors are of the form , where is any of the six units. There are two types of Eisenstein prime. First, an ordinary prime number (or "rational prime") which is congruent to is also an Eisenstein prime. Second, 3 and each rational prime congruent to are equal to the norm of an Eisentein integer . Thus, such a prime may be factored as , and these factors are Eisenstein primes: they are precisely the Eisenstein integers  whose norm is a rational prime. Euclidean domain. The ring of Eisenstein integers forms a Euclidean domain whose norm is given by the square modulus, as above: A division algorithm, applied to any dividend formula_11 and divisor formula_12, gives a quotient formula_13 and a remainder formula_14 smaller than the divisor, satisfying: Here formula_16 are all Eisenstein integers. This algorithm implies the Euclidean algorithm, which proves Euclid's lemma and the unique factorization of Eisenstein integers into Eisenstein primes. One division algorithm is as follows. First perform the division in the field of complex numbers, and write the quotient in terms  of ω: for rational formula_18. Then obtain the Eisenstein integer quotient by rounding the rational coefficients to the nearest integer: Here formula_20 may denote any of the standard rounding-to-integer functions. The reason this satisfies formula_21, while the analogous procedure fails for most other quadratic integer rings, is as follows. A fundamental domain for the ideal formula_22, acting by translations on the complex plane, is the 60°–120° rhombus with vertices formula_23. Any Eisenstein integer "α" lies inside one of the translates of this parallelogram, and the quotient formula_13 is one of its vertices. The remainder is  the square distance from "α" to this vertex, but the maximum possible distance in our algorithm is only formula_25, so formula_26. (The size of "ρ" could be slightly decreased by taking formula_13 to be the closest corner.) Quotient of by the Eisenstein integers. The quotient of the complex plane by the lattice containing all Eisenstein integers is a complex torus of real dimension 2. This is one of two tori with maximal symmetry among all such complex tori. This torus can be obtained by identifying each of the three pairs of opposite edges of a regular hexagon. (The other maximally  symmetric torus is the quotient of the complex plane by the additive lattice of Gaussian integers, and can be obtained by identifying each of the two pairs of opposite sides of a square fundamental domain, such as .)