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

160

THEORY OF EQUATIONS

Ex. 1. In Ex. 5, ~ 133, the four roots satisfy the following relations:
P 22
p2 = p24,
P3 - P2,
34
P P2,
P4 = P23.
Operate upon the left members of these equalities with the transposition
(pp2), and upon the right members with (p2p3), and show that (pp2) = (p2p3).
Ex. 2. In Ex. 1 find the transposition (ppi) which is equal to (piP3).
Ex. 3. In Ex. 1, ~ 136, find i so that (aai)- (a1a).
149. Substitutions of the Domain t2(). Since any transposition (PhPk)= (PPi), where i is some one of the numbers 0, 1, 2,
*.. (n - 1), it follows that there are not more than n distinct
transpositions in the given normal domain Q(p), which number
agrees with the degree of the domain and the degree of the
equation f(x) = 0, whose roots define this domain. Since every
number in Q(p) can be expressed as a function of p in 0t, since
every number operated upon by (ppi) passes into some other
number in the domain conjugate to it, since, moreover, no two
numbers pass into the same number (~ 147), it follows that
each such substitution applied to all the numbers in the normal
domain leaves the domain as a whole unchanged.
The substitutions (pp,), where i takes successively the values
0, 1, *.. (n- 1), are called the stbstitutions of the domain f(p).
If iV= (p) is invariant under (ppi) so that N= <(p)-=(p),
then we say that N admits of the substitution (ppi). Observe
the difference between the expressions admits and belongs to
(~ 111). In both the function must be unaltered under the
substitutions of a certain group G1, but in the latter expression
we have the additional condition that the function must be
altered by every substitution of G which does not occur in GI,
G, being regarded as a sub-group of G.
If N= +S(p) is a primitive number, then it is distinct from
each of its other conjugates ~(pl), ((p2), ', (P,,-) Hence N
admits of none of the substitutions (ppi), except, of course, the
identical substitution 1.