Kummer ring
In abstract algebra, a Kummer ring [math]\displaystyle{ \mathbb{Z}[\zeta] }[/math] is a subring of the ring of complex numbers, such that each of its elements has the form
- [math]\displaystyle{ n_0 + n_1 \zeta + n_2 \zeta^2 + ... + n_{m-1} \zeta^{m-1}\ }[/math]
where ζ is an mth root of unity, i.e.
- [math]\displaystyle{ \zeta = e^{2 \pi i / m} \ }[/math]
and n0 through nm−1 are integers.
A Kummer ring is an extension of [math]\displaystyle{ \mathbb{Z} }[/math], the ring of integers, hence the symbol [math]\displaystyle{ \mathbb{Z}[\zeta] }[/math]. Since the minimal polynomial of ζ is the mth cyclotomic polynomial, the ring [math]\displaystyle{ \mathbb{Z}[\zeta] }[/math] is an extension of degree [math]\displaystyle{ \phi(m) }[/math] (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 [math]\displaystyle{ \{1, \zeta, \zeta^2, \ldots ,\zeta^{m-1}\} }[/math]. 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
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