An introduction to the modern theory of equations, by Florian Cajori.

CYCLIC EQUATIO 'NS

199

equations of the form xm -A = 0, in which the exponents m are
the prime factors of n. By taking x'/A=z, the equation
x" — A = 0 becomes zm - 1 = 0. Hence the general solutions of
binomial equations can be given as soon as we are able to solve
binomial equations of the form z. - 1 0 whose degrees are
prime numbers. It is the latter equations which by division
by z-1 give rise to the cyclotomic equations.
Since a cyclotomic equation is a cyclic equation, its solution
is theoretically contained in ~ 172. But, as a rule, the computation of TX is extremely involved. WTe proceed to develop a
scheme, due to Gauss, by which the solution of cyclotomic
equations is divided into simpler component operations.
Ex. 1. Show that cyclotomic equations are reciprocal equations.
179. Primitive Congruence Roots. It is shown in the Theory
of Numbers that, for every prime number n, there exist numbers g (called primitive congruence roots of n), such that, on
dividing by n each member in the series,,gth2 o...t gin
the remainders obtained are (except in their sequence) the
numbers in the series
1, 2, 3,..., n-1.
For instance, if n = 5, we may take g = 2. If 2, 22, 23, 24 are
each divided by 5, the remainders are respectively 2, 4, 3, 1.
These remainders differ from the series 1, 2, 3, 4 only in the
order in which they come. Illustrate the same by taking n = 7
and g = 3.
In view of these facts and of the relation o, = 1, the roots
Co, Gil,, ow.. of the cyclotomic equation I may be written thus:
0 = ) (01= 91(9 = o),...^ (n-2 = *gn-2. This notation will
offer certain advantages. The roots of I may therefore be
written:. -0.

9 gX g,,-,. OSLc

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