Galoisförlängning

Galoisförlängning, known in English as a Galois extension, is a fundamental concept in abstract algebra, particularly in Galois theory. It describes a specific type of field extension. A field extension L over a field K is a pair of fields where K is a subfield of L. A Galois extension is a finite, separable, and normal field extension. Finite means that L is a finite-dimensional vector space over K. Separable means that the minimal polynomial of every element in L over K has distinct roots in an algebraic closure of L. Normal means that every irreducible polynomial in K[x] that has at least one root in L also splits completely into linear factors in L. The significance of Galois extensions lies in their connection to the symmetries of the roots of polynomials. The Galois group of a Galois extension L over K, which is the group of automorphisms of L that fix K pointwise, captures important information about the structure of L and its relationship to K. This group is crucial for understanding solvability of polynomial equations by radicals.