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

CHAPTER XIX

THE ALGEBRAIC SOLUTION OF EQUATIONS
190. Adjunction of Roots of Binomial Equations. In this
chapter it is proposed to develop the necessary and sufficient
conditions for the solvability of algebraic equations of any
degree. To this end we shall assume in this paragraph that
f(x) =0 is an equation which admits of being solved by algebra;
that is, we shall assume that all the roots of the given equation
f(x) = 0 can be obtained from its coefficients by a finite number
of additions, subtractions, multiplications, divisions, and extractions of roots of any index.
Let V/c, where c is an algebraic number, be any one of the
radicals which enter into the expressions for the roots of
G2
a, C,...**, a,_ of the equation f(x) =0. Thus, if c= - + II3
and m =2, then Wc is one of the radicals appearing in the
solution of the cubic, ~ 59. If c = - + + H3, m =3, we
have another radical entering the expression of the roots of a
cubic. Now the 'mth power of any radical V/c is a number in
the domain D(2,). In other words, every radical is a root of a
binomial equation of the form x -- a = 0. Thus it is evident
that all the radicals which go to make 1up a root of (x) = 0 are
roots of binomial equations.
If f(x) = 0 is reducible in the domain f2, defined by its coefficients, we may apply to its irreducible factors the argument
which follows. If f(x) = 0 is irreducible in that domain, it is
219