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.

References

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

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Kummer ring

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References