Galoisteori

Galoisteori is a branch of abstract algebra that links the theory of fields with the theory of groups. It was developed by Évariste Galois in the 19th century to address the question of which polynomial equations are solvable by radicals. The core idea is to associate a group, known as the Galois group, to each field extension.

A field extension is created when a larger field contains a smaller field. For example, the field of complex numbers contains the field of real numbers. The Galois group of a field extension captures the symmetries of the extension. Specifically, it is the group of automorphisms of the larger field that fix every element of the smaller field. These automorphisms are structure-preserving transformations that map the field onto itself.

Galois theory establishes a fundamental correspondence between the subfields of an extension and the subgroups of its Galois group. This correspondence is one-to-one and order-reversing. A key result, the fundamental theorem of Galois theory, states that this correspondence is an isomorphism of lattices. This means that relationships between subfields (like containment) are mirrored by relationships between the corresponding subgroups (like containment).

The power of Galois theory lies in its ability to translate problems about field extensions into problems about groups, which are often easier to solve. For instance, a polynomial equation is solvable by radicals if and only if its corresponding Galois group is a solvable group. This group-theoretic criterion provided a definitive answer to a long-standing problem in mathematics. Beyond solvability of polynomials, Galois theory has applications in understanding the construction of regular polygons, the transcendence of numbers, and various areas of modern algebra and number theory.