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Please help me with this question:

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 $2^2$.

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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.

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