Kummer ring

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 "m"^th root of unity, i.e.:$zeta\; =\; e^\{2\; pi\; i\; /\; m\}$and "n"₀ through "n"_"m"-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 "m"-th 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 E.E. Kummer, who studied the unique factorization of their elements.

ee also

* Gaussian integer
* Eisenstein integer
* Kummer theory