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

CHAPTER XVI

THE SOLUTION OF EQUATIONS VIEWED FROM THE STANDPOINT OF THE GALOIS THEORY
167. General Plan. Quadratic Equation. The problem, to
solve an algebraic equation, is replaced in the Galois theory
by another problem, to bring about a reduction of the Galois
group and a lowering of the degree of the Galois resolvent by
the successive adjunction of simple algebraic numbers. If a
function M is adjoined to X, the Galois group is reduced to Q.
It becomes necessary to determine the numerical value of M
for the given equation f(x) = 0. This we endeavor to do by
the construction and solution of an auxiliary equation of the
degree j, where j is the index of Q under P. The roots of
this auxiliary equation, or resolvent, are the required values of
the conjugates of M. This same process is repeated upon the
reduced Galois group until this group finally becomes 1. Then
the enlarged domain contains the roots of the given equation,
and the values of the roots may be found in terms of the
numbers Mil M',... which have been adjoined to the original
domain.
Qaadratic Equation. The Galois group of x2+ a- c + a2 = 0
is the symmetric group G2(2), ~ 158. Its only sub-group is 1,
~ 104, whose index j=2. Take M= a - al Its other conjugate value is M = a, - ca. M and M, are roots of the equation
y.2 aal - a,2 —as —4 as, ~ 161. 2 2We get y -- 2
yt2= -2aa1+2=a,2-4a2, ~ 161. We get y=~Va2-4a2
as the values of M and M,. After adjoining 31, the Galois
group is 1; the enlarged domain is 02,,1, a,, VaY-4a. We know
that a + ac = - al and a - ax = V/a2 - 4 a2. Hence
2 a = - al + /a2 - 4 a and 2 = - ac -- /a - 4 a2.
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