Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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..

share|improve this question
    
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

1 Answer 1

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 $$a=\prod_{i=1}^n[\phi_i(X)]^{m_i},\qquad b=\prod_{i=1}^n[\phi_i(X)]^{n_i-m_i},\qquad\text{and}\qquad c=1.$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.