Kummer ring

In abstract algebra, a Kummer ring $\mathbb{\mathbb{Z}}[\zeta ]$ 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 $\mathbb{\mathbb{Z}}$, the ring of integers, hence the symbol $\mathbb{\mathbb{Z}}[\zeta ]$. Since the minimal polynomial of ζ is the mth cyclotomic polynomial, the ring $\mathbb{\mathbb{Z}}[\zeta ]$ is an extension of degree $\varphi (m)$ (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 $\{1,\zeta ,{\zeta }^2,\dots ,{\zeta }^{m-1}\}$. 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