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

CHAPTER XIV

NORMAL DOMAINS
140. Theorem. A primitive number of a n.ormal domain of
the nth degree is a root of a normal equation of the nth degree.
If a number p be adjoined to 2, making Q2p) a domain of the
nth degree, every number NV in the domain 52(p) is a root of an
equation F(x) = 0 of the nth degree in 2Q, the other roots of
which are, by ~ 136, the remaining numbers conjugate to V, viz.
VN, 2Y, *.., V 1.
Since N is assumed to be primitive, F(x) = 0 is irreducible
(~ 138).
Any number Ni, being defined by ((pi), belongs to the domain
9(p). Since Q(pj is normal, we have Q2(p)-( = = = =(P -1)
(~ 132). Hence all the numbers iV, NV,..., N_ belong to the
domain Q(p), and can be expressed as functions in lQ of the
primitive number N (~ 137). From this it follows that F(x) = 0
is a normal equation.
141. Theorem. Conversely, if p is a iroot of a normal equation, then Q(p) is a normal domain of the same degree as that of
the equation.
Let po be the root, of which the other roots are functions in
Q; that is, let p, = 0((po), where v may be 1, 2, *.., or (n - 1).
Since po is a root of the given irreducible equation of the nth
degree, the domain Q(p) and all the domains conjugate to it are
of the nth degree (~ 132).
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