Symmetry and Galois Theory

Abstract

It is said that if numbers measure the physical magnitudes of objects (lengths, volumes, masses, ...), then groups measure the degrees of symmetry of geometric figures. Namely, to any figure F, we associate a group called the symmetry group of this figure, i.e., the geometric transformations f that leave the figure globally invariant, i.e., an isometry:

f(F) = F

and which preserves the distance d(f(p), f(q)) = d(p, q) for any pair of points (p, q) belonging to the figure F. A figure is said to be asymmetric if its symmetry group is the trivial group (ide), whereas, for example, the symmetry group of a regular polygon with ru sides is the dihedral group D2n (n rotations + n reflections), and the symmetry group of a circle is of infinite order (the circle has an infinite number of rotations around its center that leave it globally invariant).

Another point of view of symmetry, called arithmetic symmetry, is the study of solvability by radicals, i.e., the expressibility of the roots of the equation

anx ++ ax + a = 0

where the coefficients a, are generally in the field of rational numbers Q, using the operators (+,-,x, V, plus the coefficients of the equation.

It is shown (Galois theory) that the equation in question is solvable by radicals if and only if the permutation group of its roots is solvable in the sense of group theory.

There is a chain of subgroups from (e) up to G (the Galois group of the roots), each one normal in the next, (e) 4N, 4N2 4 4 N = G, and each quotient along the chain is abelian. Ni N G

In this Master's thesis work in Algebra and Discrete Mathematics, we aim to clarify, through examples, the meaning of solvability by radicals of an equation of degree n