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

CYCLIC EQUATIONS

195

The cubic factors yield cyclic equations of prime degree. The expression for y, selected in this example, is somewhat unwieldy. A better
choice is made in the periods of ~ 180.
Ex. 3. If m is odd, and equal to 2 n + 1, show that (- = 0,
z-1
when z + - = x, yields the cyclic equation
0 = + X — 1 - ( 1)Xn-2 - -2)- + ( - - 2)(n -3) x -4
1 2
d- (- 3)(n - 4) x_,..
1.2w
2xr
which has the roots a = 2 cos ka, where a =, and where k takes
2 n + 1
successively the values 1, 2, 3,..., n. When 2 n + 1 is prime, the equation is irreducible.
175. Theorem. Every function in Q of the roots of an irreducible cyclic equation is itself the root of a cyclic e(qlation.
Let a be a root of the given irreducible cyclic equation and
g(a) the function. Then if the values
g(-),.q( (-)), g (2(~),., g(" (~)) I
are not all distinct, let say g(oa) = g((k(a)), and we have, ~ 138,
the rectangle
gSa), g9 ((r (a), ~~,
g(~), /(~( ())..., g(-cl()),
in which the values in each column are equal, while the values
in each row are distinct, and are roots of an irreducible equa.
tion in 0, viz.,
h () m (y - g(a))))(y - ))).. (y - g(k-l())) = 0.
The consideration, as in ~ 142, of the function
S- ( F/ 7 fr. (a".)) + g (()),((a(k -2))
) L - ti(a) ~ y - /(1) - g(+. ) +J?/ (~O f/- yta) 9 - gtk 2)