Kummer ring

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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 n₀ through n_m-1 are integers.

Such Kummer ring is an extension of $\mathbb{Z}$ , the ring of integers, hence the symbol $\mathbb{Z}(\zeta)$ . It is an extension of the m^th degree, i.e. $[\mathbb{Z}(\zeta):\mathbb{Z}] = m$ .

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 of m^th degree contains

$\{1, \zeta, \zeta^2, ... ,\zeta^{m-1}\} \$ .

Such units are, by Dirichlet's unit theorem the only elements of the ring which have multiplicative inverses only in small cases; in fact for m = 5 and m ≥ 7 there are units of infinite order.

Kummer rings are named after E.E. Kummer, who studied the unique factorization of their elements.