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Let $F[X]$ be a ring of polynomials over a field $F$ and below is the uniquess part of the proof for 'Unique factoization theorem for polynomials'.

Let $\Pi_{i=1}^k [\phi_i(X)]^{n_i} = \Pi_{i=1}^k [\phi_i(X)]^{m_i}$ where $\phi_i(X)$ are irreducible monic polynomials and $n_i,m_i$ are positive integers.

The proof i have claims that if $n_i>m_i$, then $[\phi(X)]^{m_i}$ can be cancelled from both sides. How? I cannot believe this..

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What is written above is very far from a complete uniqueness proof, e.g. there is no mention or use of the key property - that irreducibles are prime. It would be better to include the complete proof, and point out which parts are not clear. – Math Gems Feb 8 '13 at 22:11
up vote 1 down vote accepted

Let $R$ be any integral domain, and suppose $ab=ac$ where $a\neq 0$. Then $a(b-c)=0$, and because $R$ is an integral domain and $a\neq 0$, we must have $(b-c)=0$, i.e. $b=c$.

Now note that $F[X]$ is an integral domain for any field $F$, and let $$a=\prod_{i=1}^n[\phi_i(X)]^{m_i},\qquad b=\prod_{i=1}^n[\phi_i(X)]^{n_i-m_i},\qquad\text{and}\qquad c=1.$$

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