Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

240 THE LINEAR GROUPS IN THREE VARIABLES [CH. XII

2~. Any positive or negative rational power of a root of
unity is again a root of unity.
3~. If n is the index of a root a, and n a positive integer,
the index of am is n/d, where d is the highest common factor
of n and m.
4~. If the index of a root 0 is n=ab, where a and b are two
integers which are prime to each other, then it is possible to
find a root of index a, say a, and one of index b, say 3, such that
0 =ao.
As is customary, we write cw, w2 for the roots of index 3;
i, -i for the roots of index 4; — w, - W2 for the roots of index
6, etc.
5~. If a is a primitive nth root, then the n roots of - 1 = 0
are a, a2,... e,, and we have
l+a+at2. +an-l=0.
6~. Theorem of Kronecker. For the proper handling of a
certain class of equations we use a very effective theorem of
Kronecker.* Instead of making a formal statement of the
theorem we shall explain its meaning by implication.
The class of equations referred to are all of the form
- - =O0; ai,..., ak being roots of unity, and the question involved is this: if these roots are not known originally,
but their number k is known, what can be inferred about their
values? The theorem implies that the k roots fall into sets,
each containing a prime number of roots the sum of which
equals zero. Moreover, if p be the number of roots in any
one of the sets, and if a be a root of index p, then the roots of
the set are E, ca,..., ap-1 where e is an unknown root of
unity. We shall discuss in full the cases k =3, 4, 5.
k=3: al +a2+a03=0. Here we have a2 =calw, a3=aico2.
k =4: al+a2-+a3 +a4 = 0. We have two sets of two roots
each, say ail+a2 = 0, a3+a4 = 0.
k=5: all+a2+-a3+a4+qa5=0. There are two possibilities:
one set only, or two sets containing 3 and 2 roots respectively.
* Memoires sur les facteurs irreductibles de l'expression x- -, Journal de
Mathematiques pures et appliquees, ser. 1, t. 19 (1854), p. 178.