# Polynomials on $\Bbb Z/p^n\Bbb Z$

Let $$n\in \Bbb N$$, let $$p$$ be a prime number and let $$\Bbb Z/ p^n\Bbb Z$$ denote the ring of integers modulo $$p^n$$ under addition and multiplication modulo $$p^n$$. Let $$f(x)$$ and $$g(x)$$ be polynomials with coefficients from the ring $$\Bbb Z/p^n\Bbb Z$$ such that $$f(x) · g(x) = 0$$. Prove that $$a_i b_j= 0$$ for all $$i,j$$ where $$a_i$$ and $$b_j$$ are the coefficients of $$f$$ and $$g$$ respectively.

I thought that it was obvious at first, because I thought each of the coefficients of $$f(x)g(x)$$ were $$a_ib_j$$, but after some thought, I observed that coefficient may be of the form $$a_1b_k+\ldots+a_kb_1$$, and $$f(x)g(x)=0$$ only guarantees $$a_1b_k+\ldots+a_kb_1$$ is divisible by $$p^n$$. Then I thought that, maybe if $$a_1b_k+\ldots+a_kb_1$$ is divisible by $$p^n$$, then each term is divisible as well. But, $$2+2$$ is divisible by $$2^2$$ and yet 2 is not divisible by $$2^2$$.

In your problem write choose representatives of $$f$$ and $$g$$ from $$\mathbb Z[X]$$ say $$F$$ and $$G$$

Then $$F=p^k F_1$$ and $$G=p^l G_1$$ where $$F_1$$ has a coefficient not divisible by $$p$$ and so does $$G_1$$

Thus $$F_1G_1$$ has a coefficient not divisible by $$p$$ but we have $$p^{k+l} f_1g_1=0$$ in $$\mathbb Z_{p^n} [X]$$

So we get $$k+l \geq n$$

Conclusion follows.