The Relationship Between Normal Extensions and Splitting Fields

Definitions and Preliminary Concepts

In field theory, a normal extension is an algebraic field extension $ L/K $ where every irreducible polynomial over $ K $ that has a root in $ L $ splits completely in $ L $. This means that $ L $ contains all the roots of every irreducible polynomial over $ K $, ensuring the extension is "normal" in the sense of having all roots in the extension. A splitting field is an extension $ L/K $ that contains all roots of some polynomial $ f(x) \in K[x] $, such that $ f(x) $ splits into linear factors in $ L $.

The study of normal extensions and splitting fields is central to understanding the structure of field extensions, particularly in the context of Galois theory. A key question arises: under what conditions is a normal extension also a splitting field? This relationship is crucial for determining the nature of field extensions and their connections to Galois groups.

Properties of Normal Extensions

A normal extension is characterized by its ability to contain all roots of any irreducible polynomial over the base field. For example, consider the extension $ \mathbb{Q}(\sqrt{2})/\mathbb{Q} $. This extension is normal because the polynomial $ x^2 - 2 $ splits completely in $ \mathbb{Q}(\sqrt{2}) $, and any irreducible polynomial over $ \mathbb{Q} $ that has a root in $ \mathbb{Q}(\sqrt{2}) $ will split entirely in the same extension. However, $ \mathbb{Q}(\sqrt{2}) $ is not a splitting field of any polynomial unless it is specifically constructed to contain all roots of a given polynomial.

The property of being normal is closely tied to the concept of separability. In a normal extension, all polynomials are separable, meaning that their derivatives are not zero in the extension. This ensures that the extension is "complete" in terms of root-finding, but it does not necessarily guarantee that the extension is a splitting field.

Properties of Splitting Fields

A splitting field is an extension that contains all roots of a given polynomial, making it a "complete" extension for that polynomial. For instance, the extension $ \mathbb{C}/\mathbb{R} $ is a splitting field for the polynomial $ x^2 - 2 $,