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

180

THEORY OF EQUATIONS

165. Adjunction of Any Irrationality. If by the acd.jnction
of any irrationlC X to 2 ice obtain a (domain S(X) in wUhich the
Galois resolvent g(y) = 0 becomes a reducible equation, so that
s1(y, X) - ( )(? - pi) *- (y - Pq-i)
is an irreducible factor of g(y) in 0(x) of the degree q, then in this
new domain the Galois group is reduced to the sub-group
1, (PPl), *", (pPq-i).
Adjoin X. Since g(y)= 0 is a normal equation in Q, ~ 146,
we have p, = c(p). In
s1(y, X) ( - p)( - pl) *.. (y - p-1) = 0 I
write &~(y) in place of y; we obtain a new equation in y, viz.,
gl(oi(y), X) = (i(y) - p)(Oi(y) - p,) '.. (4i(y) - Pq,-1)= O. II
As I is irreducible in 02 and I and II have a root p in conmmon, all the roots of I satisfy II. Let p, be any root of I; then
putting y = Ph one of the factors in II must vanish; say, the
factor 4i(Ph) - Pk
We have now the relations
Pi 0i(P)l
Pi = fs(p).
Pk -= i(ph)Hence the equality of the substitutions
(pi~k) = (pph).
Multiplying by (ppi), we have
(Ppi) (iPk) = (PPi) (PPh),
or (PPk) = (PPi)(pph).
That is, the product of any two substitutions in the set
1, (PPi),.', (PPq-.) is equal to one of the substitutions in the set.
Hence they form a group, ~ 95. Call this sub-group Q.