Let $k$ be a field. The following conditions are equivalent:
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1. The field $k$ is algebraically closed; that is, every nonconstant polynomial in $k[x]$ has a root in $k$.
2. For every nonconstant polynomial $f \in k[x]$, there exist an integer $n \geq 1$, a scalar $c \in k^\times$, and elements $a_1,\dots,a_n \in k$ such that
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\begin{align*}
f = c\prod_{j=1}^{n}(x-a_j)
\end{align*}
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in $k[x]$.
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3. Every irreducible polynomial in $k[x]$ has degree $1$.
4. Every finite algebraic [field extension](/page/Field%20Extension) $K/k$ satisfies $K=k$.