 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 ...
Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
Ferdinand Gotthold Max Eisenstein (April 16, 1823 - October 11, 1852) was a German mathematician. ...
In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
 where and a and b are integers and The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
 is a complex cube root of unity. 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. ...
The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field Q(√−3). They also form a Euclidean domain. 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 (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields...
In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ...
To see that the Eisenstein integers are algebraic integers note that each z = a + bω is a root of the monic polynomial In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...
- z2 − (2a − b)z + (a2 − ab + b2).
In particular, ω satisfies the equation - ω2 + ω + 1 = 0.
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; an Eisenstein integer x is said to be an Eisenstein prime if its only divisors are 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 consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
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. ...
 (except that we do not consider ±1, ±ω or ±ω2 themselves to be Eisenstein primes — they are units in the ring of integers). 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...
Relation to primes of the form x2 − xy + y2 It may be shown that a prime of the form x2 − xy + y2 may be factored into (x + ωy)(x + ω2y) and is therefore not prime in the Eisenstein integers. Also, a number of the form x2 − xy + y2 is prime iff x + ωy is an Eisenstein prime. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...
Euclidean domain The ring of Eisenstein integers forms a Euclidean domain whose norm v is In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ...
- v(a + ωb) = a2 − ab + b2.
This can be derived by embedding the Eisenstein integers in the complex numbers: since - v(a + ib) = a2 + b2
and since  it follows that  -
-
 .
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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