prove that $\overline{x}$ is nilpotent

Question

Let $R$ be a communtative ring, and $f(x)$ be a monic polynomial in $R[x]$

Use bar notation to denote passage to quotient ring $R[x]/(f(x))$

If $f(x) = x^n -a$ for some nilpotent element $a \in R$

prove that $\overline{x}$ is nilpotent in $R[x]/(f(x))$

I know it is sufficient to show that $x^m \in (f(x))$ for some positive $m$

But I'm really stuck trying to show this.

Any help or insight is deeply appreciated.

Answer 1

Hint: Let $k$ be a positive integer such that $a^k = 0$. In the quotient, we have $\overline{x}^n = \overline{a}$. Does this suggest an exponent you might raise $\overline{x}$ to?

Try $\overline{x}^{nk}$.