Kummer ring

In abstract algebra, a Kummer ring ℤ[ζ] 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.

A Kummer ring is an extension of ℤ, the ring of integers, hence the symbol ℤ[ζ]. Since the minimal polynomial of ζ is the m-th cyclotomic polynomial, the ring ℤ[ζ] is an extension of degree ϕ(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, ζ, ζ², …, ζ^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 E.E. Kummer, who studied the unique factorization of their elements.

See also

• Gaussian integer
• Eisenstein integer
• Kummer theory

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