GaloisGrupp

GaloisGrupp is a concept in abstract algebra that plays a crucial role in understanding field extensions and solving polynomial equations. Specifically, it is the group of automorphisms of a field extension that fix a base field.

Let K be a field and L be a field extension of K. An automorphism of L is a bijective map from L to itself that preserves addition and multiplication. A K-automorphism is an automorphism of L that fixes every element of K. The set of all K-automorphisms of L forms a group under the operation of composition, and this group is called the Galois group of the extension L/K, often denoted as Gal(L/K).

The significance of the Galois group lies in the fundamental theorem of Galois theory. This theorem establishes a one-to-one correspondence between the intermediate fields of a Galois extension and the subgroups of its Galois group. A field extension L/K is called a Galois extension if it is both normal and separable. This correspondence provides a powerful tool for studying the structure of field extensions and, consequently, for analyzing the solvability of polynomial equations by radicals.

The Galois group of a polynomial is defined as the Galois group of its splitting field over the field of coefficients. The structure of this Galois group reveals important information about the polynomial, such as whether its roots can be expressed using radicals. For example, the Abel-Ruffini theorem, which states that general polynomial equations of degree five or higher cannot be solved by radicals, can be understood through the non-solvability of the corresponding symmetric groups, which are Galois groups of certain cyclotomic fields.