Splitting field

Definition

Let $K$ be a field and let $f \in K[x]$ be nonzero. A splitting field of $f$ over $K$ is a field extension $L/K$ with these two properties.

1. The polynomial $f$ splits over $L$, as $f = c \, (x - a_1) \cdots (x - a_n)$.
2. The field $L$ is generated over $K$ by the roots: $L = K(a_1, \dots, a_n)$.

Remarks

Splitting fields exist and are unique up to isomorphism. The splitting field of $x^2 + 1$ over $\mathbb{R}$ is $\mathbb{C}$. The splitting field of $x^3 - 2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt[3]{2}, \omega)$ with $\omega = e^{2\pi i/3}$. Its degree over $\mathbb{Q}$ is $6$. The splitting field of $x^q - x$ over $\mathbb{F}_p$, for $q = p^n$, is the finite field with $q$ elements.

Explanations

L1 human

A splitting field is the smallest field that contains all the roots of a polynomial. It is where the polynomial factors completely. For $x^3 - 2$, the real cube root of $2$ is not enough, because the other two roots are not real.