Definition and Equivalence Conditions of Normal Extension

Introduction

In algebraic field theory, a normal extension is a fundamental concept that bridges the gap between algebraic closures and separable extensions. A normal extension is an algebraic field extension $ L/K $ satisfying two key properties: it is finite (i.e., $ [L:K] < \infty $) and separable (i.e., every element of $ L $ is separable over $ K $). Additionally, a normal extension is defined as a splitting field of some polynomial over $ K $. Such extensions play a crucial role in the study of field extensions, particularly in the classification of finite extensions and the resolution of polynomials. This article explores the formal definition of a normal extension and establishes the equivalence conditions that ensure its validity, emphasizing the interplay between separability, finiteness, and the existence of a splitting field.

Definition of Normal Extension

A finite extension $ L/K $ is termed a normal extension if it is a splitting field of some polynomial $ f(x) \in K[x] $. A splitting field of $ f(x) $ is a field extension $ L $ such that $ f(x) $ factors completely into linear factors over $ L $, and $ L $ is generated by the roots of $ f(x) $ over $ K $. For $ L/K $ to be a normal extension, it must satisfy the following:

  1. Separability: The extension $ L/K $ is separable, meaning that every element of $ L $ is separable over $ K $. This ensures that the minimal polynomial of any element in $ L $ over $ K $ has no repeated roots.
  2. Splitting Field: $ L $ is the splitting field of some polynomial $ f(x) \in K[x] $, implying that $ L $ contains all roots of $ f(x) $ and is generated by these roots over $ K $.

The combination of these properties ensures that $ L/K $ is a normal extension, as it is both finite and separable, and its roots are all contained within $ L $. Such extensions are particularly important in the study of Galois theory, where they are closely related to Galois groups and the solvability of polynomials.

Equivalence Conditions

The equivalence conditions for a normal extension are deeply intertwined with the properties of separability and the structure of the extension. A finite extension $ L/K $ is normal if and only if it is separable and the extension is a splitting field of some polynomial. This dual characterization provides a rigorous framework for analyzing normal extensions.

  1. Separability and Splitting Field: For $ L/K $ to be normal, it must be separable, and the splitting field of some polynomial $ f(x) \in K[x] $ must coincide with $ L $. This ensures that $ L $ contains all roots of $ f(x) $ and is generated by them over $ K $, making $ L/K $ a splitting field.
  2. Finite and Algebraic Extensions: The finiteness of $ L/K $ is essential for the extension to be a splitting field. Finite extensions guarantee that the splitting field is also finite, as the roots of any polynomial in $ K[x] $ are contained