 Eisenstein integers as intersection points of a triangular lattice in the complex plane In mathematics, Eisenstein integers, named after Ferdinand Eisenstein, are complex numbers of the form Image File history File links lattice of algebraic integers in , also called Eisenstein integers created by user:gunther using xfig File links The following pages link to this file: Eisenstein integer ...
Image File history File links lattice of algebraic integers in , also called Eisenstein integers created by user:gunther using xfig File links The following pages link to this file: Eisenstein integer ...
Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Ferdinand Gotthold Max Eisenstein (April 16, 1823 - October 11, 1852) was a German mathematician. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
 where a and b are integers and The integers are commonly denoted by the above symbol. ...
 is a complex cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane. Contrast with the Gaussian integers which form a square lattice in the complex plane. In mathematics, the nth roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ...
Triangular tiling. ...
In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...
A Gaussian integer is a complex number whose real and imaginary part are both integers. ...
Properties
The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field Q(ω). To see that the Eisenstein integers are algebraic integers note that each z = a + bω is a root of the monic polynomial In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...
In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ...
In mathematics, an algebraic number field (or simply number field) is a finite-dimensional (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension, or degree, when considered as a vector space over Q. The study of...
In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...
 In particular, ω satisfies the equation  The norm of a Eisenstein integer is just the square of its absolute value and is given by In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...
 Thus the norm of an Eisenstein integer is always an ordinary (rational) integer.
The group of units in the ring of Eisenstein integers is the cyclic group formed by the sixth roots of unity in the complex plane. Specifically, they are In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...
In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ...
- {±1, ±ω or ±ω2}
These are just the Eisenstein integers with norm one.
Eisenstein primes If x and y are Eisenstein integers, we say that x divides y if there is some Eisenstein integer z such that y = z x.
This extends the notion of divisibility for ordinary integers. Therefore we may also extend the notion of primality; a non-unit Eisenstein integer x is said to be an Eisenstein prime if its only divisors are of the form ux and u where u is any of the six units. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...
The integers are commonly denoted by the above symbol. ...
In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are 1 and itself. ...
An Eisenstein prime is an Eisenstein integer aω + b that has only two Eisenstein divisors, the complex cube root of unity and aω + b itself. ...
It may be shown that the an ordinary prime number (or rational prime) which is 3 or congruent to 1 mod 3 is of the form x2−xy+ y2 for some integers x,y and may be therefore factored into (x+ωy)(x+ω2y) and because of that it is not prime in the Eisenstein integers. Ordinary primes congruent to 2 mod 3 cannot be factored in this way and they are primes in the Eisenstein integers as well. Also, a number of the form x2−xy+y2 is rational prime if and only if x + ωy is an Eisenstein prime. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...
Euclidean domain The ring of Eisenstein integers forms a Euclidean domain whose norm N is given by In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ...
 This can be derived as follows: 
See also An Eisenstein prime is an Eisenstein integer aω + b that has only two Eisenstein divisors, the complex cube root of unity and aω + b itself. ...
A Gaussian integer is a complex number whose real and imaginary part are both integers. ...
In abstract algebra, a Kummer ring is a subring of the ring of complex numbers, such that each of its elements has the form where ζ is an mth root of unity, i. ...
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