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

~ 160] EQUATIONS WITH A REGULAR CYCLIC GROUP 307

We shall now prove this with the restriction that the domain contains an imaginary pth root e of unity. This restriction will
be removed in ~ 164 by proving that e can be found by root
extractions.
THEOREM. An equation, whose group G for a domain R
containing an imaginary pth root e of unity is a regular cyclic
group of prime order p, is solvable by radicals relatively to R.
Let xo, xi,..., Xp_ be the roots of the equation and let
G be generated by the substitution s= (xoxl... Xp). Then
s replaces the function
O - XO + iX 1 '+ 2iX2+. + e(p- l)X- 1
with coefficients in R, by e-iO. Let O = 0p. Then 0f is
unaltered by s and is therefore in R. Thus 0i is one of the
pth roots i"O of a. quantity in R. Also the sum of the roots
is given by a coefficient of our equation. Thus
Xo+X +X2 +... +Xp-l=C,
Xo+EXl1+E 2X2 +...~ -eP-1p-1=vl,
Xo+E2X 1+E 4X2 + +. 2(P- 1)Xp- 1 /2,
Xo+ ep-xl + 2(p- l)X2 +... +E(P-)(P —1) -1 -= to.
Multiply these equations by 1, e-j, -2... -(p-)
respectively, add and apply
l+E+t+2... +E(P-l=0 (t=1,..., p-).
Dividing the resulting equation by p, we get Lagrange's formulas
xi=lc+e l+ -2J e2 +. +e-(p-l)j }
(j=o, 1,..., p-i).
Since the x's are distinct by hypothesis, the O's are not
all zero. Thus a certain Oi is not zero. Since ei may be taken
as a new E, we may set 0170. While the first radical may