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

CYCLIC EQUATIONS

203

an equation of the first degree and second degree. Hence there
is a one-to-one correspondence between constructions by ruler
and compasses and algebraic operations which are purely rational
or involve square roots.
Consequently, if we wish to show the impossibility of constructing a quantity by ruler and compasses, we need only
show that the algebraic expression for that quantity in terms
of the known quantities cannot be given by a finite number of
square roots.
Applying these ideas to the problem of dividing the circle
into n equal parts by means of ruler and compasses, the problem is possible or impossible according as the roots of n_ - 1 = 0
can be expressed by a finite number of square roots or not.
If n is a prime number of the form 2k + 1, the degree n - 1
of the cyclotomic equation is a power of 2, and the operations
called for in ~ 182 involve square roots only. Hence, when n
is a prime of the form 2 + 1, the divisioa of the circle into n
equal parts by r nler tad compasses is always possible. This
important result is due to Gauss.
Ex. 1. Solve x5 — 1 = 0 by Gauss's method.
The cyclotomic equation is x4+x3+x2++1=-0. Here n-1=4=2 2;
e = 2, f = 2. It is only necessary to solve two quadratics. By trial we
get for n = 5, g = 2 the roots, cog, OWg', Cgg3
these yield the two periods
71 =t + Wg2 = + W 4,
71- = Cq + C- s3 = C2 -+ c3.
Hence equation IV, ~ 180, becomes
X2 (7- 7+ 1) + 71 = 0.
But + 711 = -C + 02 + c3 + 4 =-,
and ra71 = (co + WI) (W2 + co3)= W3 + 3 2 + W + w4 = 1.
Hence the quadratic takes the form
2 + X - =, and x -
2