# Cyclotomy

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## cyclotomy

[sī′kläd·ə·mē]*n*th roots of unity.

## Cyclotomy

the division of a circle into *n* equal parts, which is one of the oldest problems of mathematics. This problem consists in dividing a circle using only a compass and straightedge. Ancient Greek mathematicians were able to divide a circumference into 3,5, and 15 parts and to double indefinitely the number of sides of the resulting polygons. At the end of the 18th century, K. Gauss demonstrated that a circle could be divided, using a compass and straightedge, into 17 parts and into *n* parts where *n* is a prime of the form *n* = 2^{2}^{k} + 1 or a product of such primes and an arbitrary power of 2 (for *k* = 0, 1, 2, 3,4, we get the prime numbers *n* = 3, 5, 17, 257, 65, 537; for *k* = 5, 6, 7, the corresponding numbers are not prime). It is impossible, using a compass and straightedge, to divide a circle into any other number of equal parts. The problem of dividing a circle is equivalent to solving the binomial equation *x ^{n}* - 1 = 0. It is possible to divide a circle using a compass and straightedge only when the roots of this equation can be obtained by the successive solution of quadratic and linear equations.