Definition:Eisenstein Integer

Definition

An Eisenstein integer is a complex number of the form

$a + b \omega$

where $a$ and $b$ are both integers and:

$\omega = e^{2 \pi i / 3} = \dfrac 1 2 \paren {i \sqrt 3 - 1}$

that is, the (complex) cube roots of unity.

The set of all Eisenstein integers can be denoted $\Z \sqbrk \omega$:

$\Z \sqbrk \omega = \set {a + b \omega: a, b \in \Z}$

Also known as

The Eisenstein integers are also known as:

the Eisenstein-Jacobi integers, after Carl Gustav Jacob Jacobi

the Eulerian integers, after Leonhard Paul Euler.

Also see

• Eisenstein Integers form Subring of Complex Numbers

• Results about Eisenstein integers can be found here.

Source of Name

This entry was named for Ferdinand Gotthold Max Eisenstein.

Sources

• Weisstein, Eric W. "Eisenstein Integer." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EisensteinInteger.html