Abel's theorem in problems and solutions - School of Mathematics

Introduction 3
where by is indicated whichever of the defined values of the
square root. In so doing:
Going back to the coefficients we obtain
For what follows we need to recall two properties related to the equations
of second degree.
1.
Viète’s Theorem 3 : The complex numbers and are the roots of
the equation if and only if and
Indeed, if and are roots of the equation then
Eq. (3) is satisfied, from which and
Conversely, if and then, substituting and
in the equation by their expressions in terms of
and we obtain
and therefore and are roots of the equation
2.
The quadratic trinomial is a perfect square, i.e.,
for some complex number if and only if the roots of the equation
coincide (they must be both equal to This happens
if and only if (see formula (4)). The expression
is called the discriminant of the quadratic trinomial.
We consider now the reduced cubic equation
The generic equation of third degree is reduced to Eq. (5) by dividing
by After the substitution (where will be chosen later) we
obtain
3 François Viète (1540-1603) was a French mathematician.