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

NORM AL I)OMiAINS

161

150. Theorem. The substitations of the?orbmal domati 0U(p,
constitute a group of the order n.
Remembering the definition of a substitution group (~ 95),
we need only show that in the n distinct transpositions the
product of any two, say of (pp,) and (pp,), is equal to some one
of the transpositions in the set, say (ppk).
By ~ 148 we know that (ppi)= (PhPk)' Multiply both sides
by (pp,), and we get
(PPh) (PPi) = (PPb) (hPkp) = (ppk);
that is, the product of any two substitutions (pp,) and (pp) is a
substitution belonging to the set.
151. Theorem. If the equation f(x)==0 yields the Galois
domain O(p), then there corres)ponds to the group of substitutions
(pp,) of that domain a g'roup of sabstitutions si of the stom-e order
among the roots of the equations, such that the product of' Can two
su'bstitutions (ppi), (PPj) qf the domain colrrespolnds to the product
of the two correspoinding sutbstitutions si, s cof the roots off(x)= 0.
Let f(x)= 0 have the roots a, a, *.., a...,, all of them distinct. Since these roots are numbers in the Galois domain,(a, a. =- Q(P) of the degree m, it follows that
p = [(<, ***.,,s, ***.,,,_i], I
and that as = s(p) where s has any value 0, 1,..* (n-1).
Substituting for the a's their values, we get from I,
p — [(.p), '",, * (p),...,,,^-_(p)]. I
Now p is a primitive number in the Galois domaini 2(p) (~ 144),
and is, therefore, a root of the Galois resolvent g(y) = 0, whose
other roots are the remaining numbers conjugate to it, viz.
pj,., p.-1 Consider II an equation having a root p, then the
irreducible equation g(y) = 0 and the equation II have p as a
common root; hence the conjugates of p are roots common to
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