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Let $n \in \mathbb{N}$ and $p$ be a prime number. Let $f(x),g(x) \in \mathbb{Z}_{p^n}[x]$ such that, in $\mathbb{Z}_{p^n}[x]$, $f(x)g(x)=0$. Prove that $a_ib_j=0$ for all $i,j$, where $a_i$ and $b_j$ are the coefficients of $f(x)$ and $g(x)$.

I have the set of equations (for degree $r$ and degree $s$): $$a_rb_s=0, a_rb_{s-1}+a_{r-1}b_s=0,...,a_0b_0=0.$$ From these, how do I conclude $a_ib_j=0$ for all $i,j$? Also, I found that for $n=1$, the polynomial ring is an integral domain, so the conclusion holds. I had an idea to use induction on $n$, but do not know how to proceed.

user26857
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saubhik
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1 Answers1

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Hint $\ $ Clear if $f$ or $g = 0.\,$ Else $\,\color{#c00}{f = p^i \bar f},\, p\nmid \bar f\,$ and $\,\color{#0a0}{g = p^j \bar g},\ p\nmid \bar g.\,$ Gauss Lemma $\,\Rightarrow\, p\nmid \bar f\bar g.\,$

Hence $\ p^n\mid fg = p^{i+j}\bar f\bar g\,\Rightarrow\, i+j\ge n\ $ so $\ \color{#c00}{p^i\mid a_k},\ \color{#0a0}{p^j\mid b_{\ell}} \,\Rightarrow\, p^n\mid p^{i+j}\mid a_k b_{\ell}$

Remark $\ $ See also McCoy's Theorem.

Bill Dubuque
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    How is that Gauss's Lemma? In the field $\mathbb{Z}_p$, $\overline{f}$ and $\overline{g}$ are non-zero elements, hence their product is non-zero. – saubhik Feb 04 '15 at 17:14
  • @saubhik It immediately implies that the product of primitive polynomials is primitive, which is one form of Gauss's Lemma. – Bill Dubuque Mar 08 '22 at 07:23