Noun
Eisenstein integer (plural Eisenstein integers)
(algebra) A complex number of the form
a
+
b
ω
{\displaystyle a+b\omega }
, where a and b are integers and ω is defined by the following two rules: (1)
ω
3
=
1
{\displaystyle \omega ^{3}=1}
and (2)
1
+
ω
+
ω
2
=
0
{\displaystyle 1+\omega +\omega ^{2}=0}
; an element of the Euclidean domain
Z
[
ω
]
{\displaystyle \mathbb {Z} [\omega ]}
.
To divide an Eisenstein integer
a
+
b
ω
{\displaystyle a+b\omega }
by another Eisenstein integer
c
+
d
ω
{\displaystyle c+d\omega }
, notice that
(
c
+
d
ω
)
(
c
+
d
)
(
c
+
d
ω
2
)
=
c
3
+
d
3
{\displaystyle (c+d\omega )(c+d)(c+d\omega ^{2})=c^{3}+d^{3}}
; accordingly multiply both denominator and numerator (of the division expressed as a fraction) by
(
c
+
d
)
(
c
+
d
ω
2
)
{\displaystyle (c+d)(c+d\omega ^{2})}
, then simplify.