# Kummer ring

In abstract algebra, a Kummer ring ${\displaystyle \mathbb {Z} [\zeta ]}$ is a subring of the ring of complex numbers, such that each of its elements has the form

${\displaystyle n_{0}+n_{1}\zeta +n_{2}\zeta ^{2}+...+n_{m-1}\zeta ^{m-1}\ }$

where ζ is an mth root of unity, i.e.

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

and n0 through nm−1 are integers.

A Kummer ring is an extension of ${\displaystyle \mathbb {Z} }$, the ring of integers, hence the symbol ${\displaystyle \mathbb {Z} [\zeta ]}$. Since the minimal polynomial of ζ is the mth cyclotomic polynomial, the ring ${\displaystyle \mathbb {Z} [\zeta ]}$ is an extension of degree ${\displaystyle \phi (m)}$ (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 ${\displaystyle \{1,\zeta ,\zeta ^{2},\ldots ,\zeta ^{m-1}\}}$. 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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