In abstract algebra, a Kummer ring
Z
[
ζ
]
is a subring of the ring of complex numbers, such that each of its elements has the form
n
0
+
n
1
ζ
+
n
2
ζ
2
+
.
.
.
+
n
m
−
1
ζ
m
−
1
where ζ is an mth root of unity, i.e.
ζ
=
e
2
π
i
/
m
and n0 through nm−1 are integers.
A Kummer ring is an extension of
Z
, the ring of integers, hence the symbol
Z
[
ζ
]
. Since the minimal polynomial of ζ is the mth cyclotomic polynomial, the ring
Z
[
ζ
]
is an extension of degree
ϕ
(
m
)
(where φ denotes Euler's totient function).
An attempt to visualize a Kummer ring on an Argand diagram might yield something resembling a quaint Renaissance map with compass roses and rhumb lines.
The set of units of a Kummer ring contains
{
1
,
ζ
,
ζ
2
,
…
,
ζ
m
−
1
}
. By Dirichlet's unit theorem, there are also units of infinite order, except in the cases m = 1, m = 2 (in which case we have the ordinary ring of integers), the case m = 4 (the Gaussian integers) and the cases m = 3, m = 6 (the Eisenstein integers).
Kummer rings are named after Ernst Kummer, who studied the unique factorization of their elements.
