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

76

THEO1RY OF EQUATIONS

subject has been worked out mainly by C. F. Gauss, 1801, and
will be treated more fully in Chapter XVII under the head
of Cyclotomic Equations.
65. Roots of Unity. We give a few general properties of
the nth roots of unity, some of which are evident from previous considerations.
I. The equation xn = 1 has no multiple roots.
Here f(x) = x" - 1, f'(x) = nx-1. Since f(x) and f'(x) have
no common factor involving x, there are no multiple roots
(~ 21).
II. If a is a root of xn - 1 = 0, then Ck is also a root, k being
any integer.
Since an = 1, it follows that ack = 1 or (a')" = 1, where k is
zero or any integer, positive or negative. Hence ak is a root
of unity. As there are only n roots, it is evident that the
powers of a are not all distinct from each other, and ak is a
periodic function.
III. If mr and n are prime to each other, the eqnations xm - 1 = 0
and x -1. = 0 have no common root except 1.
First we prove the theorem: If nm and n are prime to each
other, then it is always possible to find integers a and b such that
mb - na = 1. The fraction m may be expanded into a tern
minating continued fraction, say
m 1
n=P+
n +
9+r
The successive convergents are p, Vq +, 2)(+ )~+~., Subq qr+ l
tracting the last but one convergent from the last, we obtain
a fraction whose numerator, pq(qr + 1) + qr - (pq + 1)(qr + 1)
is seen to be equal to - 1. (By mathematical induction it may