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

136

THEORY OF EQUATIONS

degree with respect to x, such that the coefficients of the factors are numbers belonging to the domain lQ, then the function
f(x) is called reducible in Q; otherwise it is called irreducible
in 02.
ill Q.
Thus, if Q designates the domain of rational numbers,
then x2 - y2 is reducible in Q2, because it yields the factors
(x + y) (x - y). On the other hand, x -- 3 ys is irreducible in
Q, because some of the coefficients of its factors
(x + 4V y) (x - A3 y)
are not rational.
If, however, we form a new domain by the adjunction of
a=V/3 to the domain of rational numbers, we obtain 2(,,a)
embracing numbers of the kind a + -/3 b, where a and b are
rational. With respect to this larger domain the functions
2 — y2 and x2 - 3 y2 are on an equal footing, for both are
reducible in (l,), since the coefficients of the two factors of
each function are numbers belonging to the same domain 02(, ).
Ex. 1. Find out which of the following functions are reducible in the
domain of rational numbers Q(i):
(a) 2 + 2 x +1,
(b) 4 + x2.+ 1,
(c) x2 + x -1,
(d) X2 + x + 1,
(e) x2 +1.
Ex. 2. For each of the above functions which are irreducible in (1,),
find by adjunction the smallest new domain in which the function is
reducible.
Ex. 3. Find a domain such that all the functions of Ex. 1 will be
reducible in it.
124. Algebraic Numbers. All numbers which are roots of an
algebraic equation
f(x) _ a(t0 + tx"- +... + a,_-x + a,, = 0