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If the coefficients of a polynomial p(x) are all real integers, then every root of p(x) is either 1) real or 2) a complex number whose conjugate is also a root of p(x).

Is there any easy way to separate $p(x)\rightarrow p_1(x)p_2(x)$, where $p_1$ has only real roots and $p_2$ has only complex roots? I understand that $p_1$ and $p_2$ will not generally have rational coefficients.

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