On the Solvability of Polynomial Equations

Up to this point, we have used terms such as minimal polynomial, Galois conjugacy, and Galois group for polynomials with rational coefficients. It turns out that the theory becomes much more powerful when we also consider extensions of the rational numbers as the coefficient domain. We call this view the relative, while we refer to the rational number case as the absolute. For example, \(X^4 - 2\) is the minimal polynomial of a fourth root \({\sqrt [4]{2}}\) of 2 over the rational numbers. On the other hand, \(X^2 - \sqrt {2}\) is the minimal polynomial of the same algebraic number, but this time over the number field \(\mathbf {Q}(\sqrt {2})\) , consisting of all algebraic numbers that are rational in \(\sqrt {2}\) . Accordingly, the degree of \(\sqrt [4]{2}\) over \(\mathbf {Q}\) is exactly 4, but over \(\mathbf {Q}(\sqrt {2})\) it is only 2. Just as the degree decreases, there are fewer Galois conjugates, and the Galois group becomes smaller the larger we choose the number field. We can use this observation to better understand the absolute case over the rational numbers: First, we look at the relative case over suitable extensions of the original coefficient domain. Then we successively reduce the number field, so that the Galois group successively increases until we find the Galois group over the rational numbers in the limit. Extensions of the number field correspond exactly to subgroups of the Galois group, and we can specify an exact bijective correspondence in the form of the fundamental theorem of Galois theory. With the help of this main theorem, we reduce difficult questions such as the solvability of equations to simpler questions about finite groups. After we show that the roots of the cyclotomic polynomials (and thus the vertices of an equilateral n-gon when suitably drawn in the complex number plane) are given by radical expressions, we can deduce Galois’ result, a necessary and sufficient condition for the solvability of polynomial equations by radicals. It turns out that the general equation of degree five or higher is not solvable by radicals. There also special equations of degree five and higher that are not solvable by radicals. At the end of the chapter, we derive the solution formulas for equations of degree three and four, known since the beginning of the modern era, but using Galois’ theory.