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Let $p$ be a polynomial in $\mathbb{Z}[t]$ with $p(0) = \pm 1$ and $L$ a positive integer, then there is a polynomial $q \in \mathbb{Z}[t]$ such that $p$ divides $t^Lq - 1$.

I know that if $p(t) = t - 1$, then for any integer $L$, we can take $q(t) = 1$, since we can write $$t^L -1 = (t -1)(1 + x + x^2 + \dots + x^{L-1}),$$ which shows that $p(t)$ divides $t^Lq(t) -1$.

For a more general polynomial $p$, I am not sure how to proceed.

I will be grateful for any help you can provide :)

Luísa Borsato
  • 2,316

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If $p(0)=-1$, let $p(t)=-1+tQ(t)$. Then $$(p(t)+1)^L=t^LQ(t)^L$$ and so $p(t)$ divides $t^LQ(t)^L-1$. Set $q(t)=Q(t)^L$.

If $p(0)=1$, let $p(t)=1+tQ(t)$. Then $$(p(t)+1)^{2L}=t^{2L}Q(t)^{2L}$$ and so $p(t)$ divides $t^{2L}Q(t)^{2L}-1$. Set $q(t)=t^LQ(t)^{2L}$.