Kummer ring

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In abstract algebra, a Kummer ring \mathbb{Z}[\zeta] is a subring of the ring of complex numbers, such that each of its elements has the form

 n_0 + n_1 \zeta + n_2 \zeta^2 + ... + n_{m-1} \zeta^{m-1}\

where ζ is an mth root of unity, i.e.

 \zeta = e^{2 \pi i / m} \

and n0 through nm−1 are integers.

A Kummer ring is an extension of \mathbb{Z}, the ring of integers, hence the symbol \mathbb{Z}[\zeta]. Since the minimal polynomial of ζ is the mth cyclotomic polynomial, the ring \mathbb{Z}[\zeta] is an extension of degree \phi(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, \zeta, \zeta^2, \ldots ,\zeta^{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.

See also[edit]

References[edit]

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