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

GALOIS THEORY OF ALGEBRAIC NUMBEItS 137

with integral coefficients are called algebraic members. Numbers
which cannot occur as roots of an algebraic equation are called
transcendental. It was first proved by Hermite (1873) that e,
the base of the natural system of logarithms, is a transcendental number. In 1882 Lindemann first demonstrated that
7r, the ratio of the circumference of a circle to its diameter, is
also transcendental. If to the domain of rational numbers
j(~) we adjoin 7r, we obtain a transcendental domain. If the
number adjoined to Q(,) is algebraic, the new domain is called
an algebraic domain.
125. Irreducible Equations. An equation, f(x) = 0 is said to
be reducible or irreducible in a domain 2, according as the
function f() is reducible or irreducible in 0.
If we adjoin to the domain Q one of the roots a of the
equation f() = 0, then if a does not belong to the domain 0,
we obtain a new domain (,,.) which is an algebraic domain
over 0.
126. Theorem. If f(x) = 0 and F(x) = 0 are both equations
in the domain 0, and if f(x)- 0 is irredutcible in. 0 and has one
root which sctisfies F(x) = 0, then all its roots satisfy F(x) = 0.
Since the two equations have at least one root in common,
the two functions f (x) and F (x) have a common factor involving x. But we know that the highest common factor is found
by ordinary division, i.e. by a process which nowhere introduces numbers not found in the given domain of rationality.
The highest common factor is therefore a function in fl.
Iut f(x), being irreducible, has no factor in Q involving x,
except itself. Hence the highest common factor must be either
(x) or a quantity differing from f(x) by a constant number.
[n other words, we must have either F(x) = c f(x) or
F(x) = g(x). /'(x), where g(x) is a function in Q.
Ex. 1. The cubic x3 - 2 x2 - + = 0 has three incommensurable
roots and is therefore irreducible in the domain Q(1). It has one root in