8 lines
836 B
TeX
8 lines
836 B
TeX
\section{Unique Factorization In $\mathbf{\Z[x]}$}
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\begin{theorem}[Unique Factorization in $\mathbf{\Z[x]}$]
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Every polynomial in $\Z[x]$ that is not the zero polynomial or a unit in $\Z[x]$ can be written in the form $b_1b_2\dots b_sp_1(x)p_2(x)\dots p_m(x)$, where the $b_i$'s are irreducible polynomials of degree 0 and the $p_i(x)$'s are irreducible polynomials of positive degree. Furthermore, if
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\[ b_1b_2\dots b_sp_1(x)p_2(x) \dots p_m(x) = c_1c_2 \dots c_tq_1(x) q_2(x) \dots q_n(x) \]
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\noindent where the $b_i$'s and the $c_i$'s are irreducible polynomials of degree 0 and the $p_i(x)$'s and $q_i(x)$'s are irreducible polynomials of positive degree, then $s=t, m=n$, and, after renumbering the $c$'s and $q(x)$'s, we have $b_i = \pm c_i$, for $i=1, \dots, s$, and $p_i(x)= \pm q_i(x)$, for $i = 1, \dots, m$.
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\end{theorem}
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