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where ζ is an mth root of unity, i.e.
and n0 through nm−1 are integers.
A Kummer ring is an extension of , the ring of integers, hence the symbol . Since the minimal polynomial of ζ is the mth cyclotomic polynomial, the ring is an extension of degree (where φ denotes Euler's totient function).
The set of units of a Kummer ring contains . 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).
- Allan Clark Elements of Abstract Algebra (1984 Courier Dover) p. 149