Normal extensions

Last updated January 27, 2022

Idea

A normal extension $L/K$ is one that does not live “between” irreducible polynomials— it either contains all of the roots or none of them, similar to the idea of the Prime Avoidance Lemma.

Definition

Let $K$ be an algebraic extension of $k$ contained in an algebraic closure $\overline{k}$ of $k$.

Definition. The extension $K/k$ is called normal if one of the following equivalent conditions hold:

1. All $\sigma\in\sum_{K/k}^{\overline{k}/k}$ have the same image (in particular they all induce automorphisms of $K$)
2. $K$ is the splitting field of a family of polynomials in $k[T]$.
3. Every irreducible polynomial in $k[T]$ which has a root in $K$ splits into linear factors in $K$

Properties

Let $k\subset K\subset L$ be a tower of algebraic extensions.

1. If $L/k$ is normal then $L/K$ is normal.
2. Fundamental thm of Galois Theory: $M/k$ is normal iff $H\triangleleft G$

Related

• Field extension