Kummer ring
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.e.
and n0 through nm−1 are integers.
A Kummer ring is an extension of, the ring of integers, hence the symbol. Since the minimal polynomial of ζ is the mth cyclotomic polynomial, the ring is an extension of degree .
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
By Dirichlet's unit theorem, there are also units of infinite order,
except in the cases, , the case and the cases, .
Kummer rings are named after Ernst Kummer, who studied the unique factorization of their elements.
