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Characterization of irreducible polynomials
In an old textbook, I read the following:
A polynomial $p(x) \in \mathbb{Z}[x]$ has a root in $\mathbb{Q}$ $\Longleftrightarrow$ $p(x)$ is irreducible $\Longleftrightarrow$ $p(x)$ is a product of distinct linear factors.
The latter is easy to show by finding the factors of $p$ in $\mathbb{Z}[x]$. However, my textbook states the first as a definition without any sort of proof. Could someone please provide a proof/reference for this statement? Thank you.
A:
The implication $\implies$ is easy. If $p(x)$ is irreducible, then it is the product of linear factors $p(x) = \prod_{i=1}^n(x - x_i)$ over all roots $x_i$ of $p(x)$. If $p(x) = \prod_{i=1}^n(x - x_i)$, it suffices to show $x - x_i
eq 0$ for each $i$.
Suppose $x - x_i = 0$. Then $p(x) = \prod_{i
eq j}(x - x_i) \implies \prod_{i
eq j}(x - x_i) \mid p(x) \implies p(x) = (x - x_i)\prod_{i
eq j}(x - x_i) = x^2 \cdot \prod_{i
eq j}(x - x_i)$. This contradicts $p 0b46394aab
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