Splitting field

Last updated January 27, 2022

Idea

Given a polynomial $f\in k[T]$, the splitting field $K$ of $f$ is the extension $K/k$ such that $f$ splits into linear factors into $K$ and is minimal in the sense that $K=k(\alpha_1,\dots,\alpha_n)$ is generated by all roots of $f$.

Definition

Definition. Suppose $P\in k[T]$ is of degree $\geq 1$. Then the extension $K/k$ is called its splitting field iff

1. $P=\prod_{i=1}^d(T-\alpha_i)$ in $K$
2. $K=k(\alpha_1,\dots,\alpha_d)$.

Properties

1. Let $K_1/k$, $K_2/k$ be two splitting fields for the same polynomial $P$. Then there exists an isomorphism $\sigma:K_1/k\overset{\simeq}{\to}K_2/k$.
2. If $k\subset K_2\subset \overline{k}$ then any $\sigma':K_1/k\to\overline{k}/k$ maps $K_1$ to $K_2$.
3. A splitting field always exists. Consider the field generated by the roots of $f$ in the algebraic closure of $k$.