8
votes
1answer
432 views

$f'/f\in\mathbb{Z}[[x]]$ for polynomials vs. formal power series $f$

I am curious about the following problem from MIT's Problem Solving Seminar (#26 here, though the link may stop working after a few weeks): Let $f(x) = a_0+a_1x+\cdots\in\mathbb{Z}[[x]]$ be a ...
5
votes
1answer
50 views

Kernel of the evaluation map on a power series ring

Let $R$ be a commutative ring with unity and $r \in R$ a nilpotent element. Is it true that if $f \in R[[\epsilon]]$ satisfies $f(r) = 0$, then $(\epsilon - r) | f$ in $R[[\epsilon]]$? I tried solving ...