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Concept
Eisenstein integer
Summary
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 a and b 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.
The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field — the third cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each z = a + bω is a root of the monic polynomial
In particular, ω satisfies the equation
The product of two Eisenstein integers a + bω and c + dω 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 ω satisfies
The group of units in this ring is the cyclic group formed by the sixth roots of unity in the complex plane: the Eisenstein integers of norm 1.
The ring of Eisenstein integers forms a Euclidean domain whose norm N is given by the square modulus, as above:
A division algorithm, applied to any dividend and divisor , gives a quotient
and a remainder smaller than the divisor, satisfying:
Here 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 . Then obtain the Eisenstein integer quotient by rounding the rational coefficients to the nearest integer:
Here may denote any of the standard rounding-to-integer functions.
The reason this satisfies , while the analogous procedure fails for most other quadratic integer rings, is as follows.
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