Split polynomial

Definition

Let $L$ be a field and let $f \in L[x]$ be nonzero. The polynomial $f$ splits over $L$ if there are $c, a_1, \dots, a_n \in L$ with $$f = c \, (x - a_1)(x - a_2) \cdots (x - a_n).$$

Remarks

Then $c$ is the leading coefficient of $f$, $n = \deg f$, and $a_1, \dots, a_n$ are the roots of $f$, not necessarily distinct. The polynomial $x^2 - 2$ splits over $\mathbb{R}$, but not over $\mathbb{Q}$. The polynomial $x^2 + 1$ splits over $\mathbb{C}$, but not over $\mathbb{R}$. Over an algebraically closed field, every nonzero polynomial splits.

Explanations

L1 human

A polynomial splits when you can factor it completely into linear factors. Then you can read all its roots from the factors.