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

130

THEORY OF EQUAT'IONS

If we multiply the equations in I by o-k, ol-k, *', ton_-k,
respectively, and then add, we have the more general result,
o-k [W, a] = na.k III
Hence, if we are given the values of the Lagrangian resolvents
of an equation f(x)= 0 of the nth degree and the nth roots of
unity, the equation f(x) = 0 is solved.
117. Theorem. If we operate upon the subscripts of a in
[C, a] with the cyclic suibstitultion (O 1 2 3 *.. (n - 1)), [w, a]
becomes w-1 [o, a]; if we operate with (0 1 2... (n - l))k, [Co, a]
becomes ow [ow, a].
If we operate upon
[), a] =_ a + (Oc1 + + W 01-1n
with the substitution (0 1 2 *.. (n - 1)) and observe that
o-l = (-1, etc., we get
-1 [(0, (at] ( + ( 4a2 3 + w2a,3 + + an-l,
( + + a + + *+,.0"1-, ).
Operating in this manner k times, we can easily establish the
truth of the second part of the theorem.
118. Theorem. If with the cyclic substitution
(O 1 2... ( - ))
we operate upon the subscripts of a in [o, a], the subscript of the
coefficient of each power of o in [o, a]v undergoes the cyclic substitution (0 1 2... (n- 1)),' v being any positive integer.
By the Polynomial Formula expand
[(, (a]- (a + a1l + ** + _lan-l)V
and by the relation U" = 1 reduce all exponents of w to
exponents less than n. Then combine all terms having like
powers of w. We get
[ow, al] = Ao + wA1 + 2A +.. + o"-ll, I