How to find the number of maximal ideal in the Quotient ring $\frac {\mathbb{Z}_5{[x]}}{<(x+1)^2(x+2)^3>}$. ?
$R/A$ is an integral domain iff $A$ is prime.
2026-03-30 23:12:19.1774912339
Number of maximal ideal in the quotient ring
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Note that there is a one-to-one correspondence between maximal ideals of $\Bbb Z_5[x]/(x+1)^2(x+3)^3$ and maximal ideals of $\Bbb Z_5[x]$ containing $(x+1)^2(x+3)^3$. Now $\Bbb Z_5[x]$ is a PID and every prime ideal is maximal. So if a maximal ideal $(f)$ contains the ideal $((x+1)^2(x+3)^3)$, then $f$ must be irreducible and must divide the polynomial $(x+1)^2(x+3)^3$. Therefore $f = (x+1)$ or $f=(x+3)$ and these are easily seen to be irreducible. Therefore $\Bbb Z_5[x]/(x+1)^2(x+3)^3$ has exactly two maximal ideals.