Moderate# Splitting field of $K/F$: Definition and Examples

ABSALG-MH2NPW

By the Fundamental Theorem of Algebra, every polynomial $f(x)\in {\mathbb C}[x]$ is a product of linear factors in ${\mathbb C}[x]$.

Let $F\subseteq{\mathbb C}$ be a field. If $f(x)\in F[x]$, and $f(x)=a_n(x-\alpha_1)\ldots(x-\alpha_n)$, $\alpha_1,\ldots,\alpha_n\in{\mathbb C}$, then the **splitting field over $F$ of $f$** is the field $F(\alpha_1,\ldots,\alpha_n)$ generated over $F$ by all the roots of $f(x)$.

Which of the following are true?

Select **ALL** that apply.