Theorem: Let $R$ be an integral domain, then ring of polynomials $R[x]$ form an integral domain.
To prove this, i first prove a little lemma (not giving the proof here):
Lemma: Let $R$ be integral domain, then $\forall p,q\in R[x]$ we have that $\deg{(pq)}=\deg{(p)}+\deg{(q)}$.
In order to prove the theorem, we have to show that if we have any two nonzero polynomials $p,q$, then their product is nonzero. Because $p,q$ is nonzero, we have that their degree is greater or equal to 0. By lemma $\deg{(pq)}=\deg{(p)}+\deg{(q)}\geq0+0=0$ this shows that $\deg{(pq)}\geq 0$ so $pq$ is nonzero and we conclude that $R[x]$ forms an integral domain.
Is my proof valid or should I aim for something stronger? All the proofs I am aware of always are long examining the terms of the polynomial $pq$ by definition and so on, so I was just wondering, if this argument is correct. Thanks
Best Answer
Your proof is perfectly valid.
I doubt the proofs are that much longer; all the work that goes into examining coefficients of $pq$ will instead go into proving the lemma.