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

ABELIAN EQUATIONS

217

From I we see that replacing a by f or y changes M into M,
or M2. Hence, if
ml = X(II) = F4 (a), MV2 = X1(IM) = F1 (a),
we may write
X(1I2) = F(y) = XX (XI) = F,41 (a)
X1(31) = Fib(g) = X1X (11) -= F1 (a).
Since, by assumption, q, l(cr) = D,4>(a), we have also XX1(M)
=XX(M). Similarly for other conjugates of if. We have
now proved that g(lM)= 0 is an Abelian equation.
Hence we have shown that the solution of the given Abelian
equation f(x) = 0 can be reduced to the solution of cyclic equations and of another Abelian equation of lower degree. The
latter Abelian equation can be treated in the same manner
as was f(x)= 0; hence, eventually, the solution of f(x) = 0 is
reduced to that of cyclic equations only.
Ex. 1. Abel gave the following example of an Abelian equation. Let
a -; then cos a, cos 2 a, *.., cos na can be shown to be the roots of
the equation xn x- n(n - 3) - xn-4 +.. = 0. I
4 16 1.2
For the derivation of this equation see Serret's Algebra (Ed. G.
Wertheim), 1878, Vol. I, pp. 195-199. Expanding the right member
of De Moivre's formula, cos ma + i sin ma = (cos a i sin a)m, by the
binomial theorem, we can express cos ma as a function in Q(1) of cos a.
We may, therefore, write cos ma = (cos a), where 0 is the function.
Similarly, cos mna = l1(cos a). Writing mla for a in the former equation,
we get cos (mm1a) -= (cos nma) = 0, (cos a).
If in l(cos a) = cos mna we replace a by ma, we have
cos (mnma) -= 0(cos ma) = 010(cos a).
Hence every root of I can be expressed as a function in 0 of one of
them, and we have in addition
O0,(cos C) =- 01(cos a)o
Therefore I is an Abelian equation.