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How to find the characteristic if degree of f(x) is n? $$\mathrm{char}\Bigl(\mathbb{Z}[x]/\langle f(x)\rangle\Bigr)$$

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The characteristic of a ring is the smallest number of times you can add $1$ to itself and get 0. If this never happens you have a ring of characteristic $0$, which is true of $\mathbb Z[x]$. If $f(x)$ is a constant polynomial of degree $0$, then you can change the characteristic of the quotient ring. But otherwise, going modulo a polynomial doesn't change the characteristic of the ring.

For example, if $\bar{1} + \ldots + \bar{1}=\bar{0}$ ($p$ times) in the quotient ring $\mathbb Z[x]/f$, then this means that $1+(f) + \ldots + 1+(f)=p+(f)$ is in the ideal $(f)$. This means that $p+fg=fh$, i.e. $p=f(g-h)$ which means that $f$ must be a constant which divides $p$.

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