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 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 (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


  • Allan Clark Elements of Abstract Algebra (1984 Courier Dover) p. 149
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