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

RESOLVENTS OF LAGRANGE

131

where A0, 'A,, A,,_ are expressions of the degree v with respect to a, aC, a*,..., a,_-, and have integral numerical coefficients.
If in formula I we replace (o by o(0, w2, "), (,,1 * in succession, we get the following n formule:
[<, a] = Ao + oAl + wA + A2 + * * 1 + (^-lA, 1
[oi, Cv- A, +,A1 + W12A1 +- * + +1o1-?n i II
[on_, a]v_ A0o (~ w_1A1 + (0n-1,A + ~* + ~o.-l1-A'. j
It was shown in ~ 69, Ex. 5, that the sum of the pth power of
the nth roots of unity is n or 0, according as p is divisible or
not divisible by n. Remlembering this and multiplying the n
expressions in II by o-k, wl, ***, o,_l-k, respectively (k being
any integer), we get, after adding the n resulting expressions,
nA = S-k o, oa]v, III
where S indicates the sum of all the expressions obtained by
writing in succession (, o, Io,, ***, )n-I in place of o. If now we
operate upon the subscripts of a, occurring in each of the v factors [o, a] in the right member of III with the cyclic substitution (0 1 2... n - 1), we get, ~ 117,
W)-'-. [o, a]v. IV
Now, by writing ck + v for k in formula III, we obtain
(o)-" [-, a- = fnAk+v.
In other words, the substitution (0 1 2 *.. (n- 1)), applied
to the subscripts of a in the right member of III causes Ak
to be replaced by Ak+,. But Ak is transformed directly into
Ak+^ by the application to its subscript of the substitution
(0 1 2... (n - 1))v. Hence the theorem is established.
Ex. 1. Illustrate this theorem by the roots aC, acx, a2 of the cubic,
taking v = 2.
We have [to, ao] = (co + C(w1 + 2(a2,
[, (to]2 = Ao + A1w + Asw2,
where Ao = ao2 + 2 (a1X2, Al = a22 + 2 0a1l, A2 = a12 + 2 (0Oa2.