Kummer ring
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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 n_{0} through n_{m−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 (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 . 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.
See also
References
 Allan Clark Elements of Abstract Algebra (1984 Courier Dover) p. 149