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

IRESOLVENTS OF LAGRANGE

133

To the subscripts of a in the right member of I apply thl
substitution (0 1 2 * (n - 1)), and we get
~i k -v-vl,-' [n 0, V '. [As, a]^.t.,
which expression is recognized by I to be equal to nBk+v+~,,+.....
But Bk is replaced by PBk,,++P,,,..., if we operate upon Bk with
the substitution (0 1 2... ( - 1))v +i+''-. Hence the theorem
is established.
* Ex. 1. Show that the function [w, a]n belongs to the cyclic group of
the degree n.
If we operate upon [w, (X] with any such substitution (0 1 2... (n - 1))
of the cyclic group, the effect is the same upon the coefficients Bk of
[WC, (a] as if the substitution (0 1 2... ( - 1))" were applied to the
subscripts of Bk directly, ~ 118. But (0 1 2... ( - 1))" is the identical substitution; hence it brings about no change. Consequently
[C, a]" is invariant for the cyclic group. This invariance holds for no
substitution of the symmetric group of degree n, except the substitutions
which occur also in the cyclic group. Hence [cw, a]n belongs to the cyclic
group.
* Ex. 2. Show that the product [a, a]n-. [CX, a] belongs to the
cyclic group of degree n.
By ~ 118, IV, the cyclic substitution (0 1 2.. n - 1), effected upon
the subscripts of a in [c, a]Xn- gives w-12+ [c, a]"'-. When operated
upon those in [C, (a] it gives w-t[wa, a]. Hence, when operated upon
the product of the two, we get — "+X —A[w, a]"-x. [ow, (], where
w-n+X-x.= — n - 1.
Ex. 3. Show that (a - ial - a2 + ita3)4 belongs to the cyclic group of
degree four.
For convenience, let - i =, and we have (. waX+ -- w2a2 + Wo3a3)4,
which, by ~ 118, IV, becomes w-4(a + wa,- + j'- a2 + cc3a)4 when operated
upon by (0 1 2 3).
Ex. 4. Notice if the following functions belong to the cyclic group of
degree four:
(ca- Iat - ac2 - ti(3)4,
(a - ial - a2 + is3) (a + ial - a2- ia3),
(a - 1 + t2 - "3) ((t - i 2 + )2,
(a - a1 + a2 - a3)2.