Galois group's initial definition (automorphism group of an extension field)

Introduction

In modern algebra, the concept of a Galois group emerges as a cornerstone of field theory, providing a powerful framework to study the symmetries of field extensions. A Galois extension is a field extension $ E/F $ that is both normal and separable. Normality ensures that every irreducible polynomial over $ F $ that has a root in $ E $ splits completely in $ E $, while separability guarantees that all roots of polynomials over $ F $ are distinct. This combination of properties ensures that the automorphism group of $ E $, denoted $ \text{Aut}_F(E) $, is a group under composition, and the Galois group $ \text{Gal}(E/F) $ is precisely this group.

The automorphism group $ \text{Aut}_F(E) $ consists of all field automorphisms of $ E $ that fix $ F $, i.e., maps $ F $ to itself. Such automorphisms preserve the field operations and the structure of $ E $, making them central to understanding the symmetries of the extension. The Galois group, therefore, captures the intrinsic symmetries of the extension, linking algebraic structures to group theory.

Automorphism Groups of Field Extensions

A field extension $ E/F $ is a pair of fields where $ F $ is a subfield of $ E $. The automorphism group $ \text{Aut}_F(E) $ is the set of all $ F $-linear maps $ \sigma: E \to E $ that preserve addition and multiplication. These automorphisms are bijective and satisfy $ \sigma(F) = F $, ensuring they act on the entire extension.

For finite extensions, the automorphism group is finite, and its size is equal to the degree of the extension $ [E:F] $. This property is critical in the study of Galois extensions, as it ensures that the group structure is well-defined and manageable. Moreover, the automorphism group is a group under composition, and its elements correspond to the "symmetries" of the extension.

The Galois Group as Automorphism Group

The Galois group $ \text{Gal}(E/F) $ is the automorphism group of a Galois extension $ E/F $, defined as the set of all $ F $-automorphisms of $ E $ that fix $ F $. This group is denoted $ \text{Gal}(E/F) $ and is a fundamental object in field theory.

For a Galois extension, the automorphism group is both finite and faithful, meaning that every non-trivial element corresponds to a distinct automorphism. This faithfulness ensures that the group structure is isomorphic to the group of automorphisms of the extension, providing a bijection between the group and the automorphism group. Additionally, the Galois group acts on the roots of polynomials in $ E $, and its structure is closely tied to the splitting fields and intermediate fields of $ E $.

Key Properties and Theorems

A Galois extension $ E/F $ is characterized by its automorphism group $ \text{Gal}(E/F) $, which is a group under composition. The group is also faithful and surjective, ensuring that every automorphism of $ E $ that fixes $ F $ is captured in the group. For finite extensions, the size of $ \text{Gal}(E/F) $ is