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 mth root of unity, i.e.

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

and n₀ 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 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

• Kummer theory

References

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