Splitting Fields and Normal Extensions

Splitting fields are the minimal fields in which a polynomial factors completely into linear factors. They are unique up to isomorphism and fundamental to Galois theory.

Splitting Fields

Definition7.3Splitting field

A splitting field of a polynomial $f \in F[x]$ over $F$ is a field extension $K/F$ such that:

$f$ splits completely in $K[x]$ : $f(x) = c(x - \alpha_1) \cdots (x - \alpha_n)$ with $\alpha_i \in K$ .
$K = F(\alpha_1, \ldots, \alpha_n)$ (minimality).

Theorem7.3Existence and uniqueness

For any nonconstant $f \in F[x]$ , a splitting field exists and is unique up to $F$ -isomorphism. If $\deg(f) = n$ , then $[K:F]$ divides $n!$ .

ExampleSplitting field examples

$f = x^2 - 2$ over $\mathbb{Q}$ : splitting field $\mathbb{Q}(\sqrt{2})$ , degree $2$ .
$f = x^3 - 2$ over $\mathbb{Q}$ : splitting field $\mathbb{Q}(\sqrt[3]{2}, \omega)$ where $\omega = e^{2\pi i/3}$ , degree $6$ .
$f = x^4 + 1$ over $\mathbb{Q}$ : splitting field $\mathbb{Q}(\zeta_8)$ where $\zeta_8 = e^{2\pi i/8}$ , degree $4$ .
$f = x^p - 1$ over $\mathbb{Q}$ : splitting field $\mathbb{Q}(\zeta_p)$ , degree $p - 1$ .

Normal Extensions

Definition7.4Normal extension

A field extension $K/F$ is normal if every irreducible polynomial in $F[x]$ that has one root in $K$ splits completely in $K[x]$ .

Equivalently (for finite extensions): $K$ is the splitting field of some polynomial $f \in F[x]$ .

ExampleNormal and non-normal extensions

$\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ : normal (splitting field of $x^2 - 2$ ).
$\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ : not normal. The polynomial $x^3 - 2$ has one root ( $\sqrt[3]{2}$ ) in $\mathbb{Q}(\sqrt[3]{2})$ but does not split completely there (the other roots $\sqrt[3]{2}\omega, \sqrt[3]{2}\omega^2$ are not real).
$\mathbb{Q}(\sqrt[3]{2}, \omega)/\mathbb{Q}$ : normal (the splitting field of $x^3 - 2$ ).

RemarkNormal closure

Every finite extension $K/F$ has a normal closure: the smallest normal extension $N/F$ containing $K$ . If $K = F(\alpha)$ with minimal polynomial $m$ , then $N$ is the splitting field of $m$ over $F$ .