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Number of maximal ideals in the ring $\mathbb{Z}_5[x]/\langle (x+1)^2(x+2)^3 \rangle$ is

$(a)$ infinite

$(b)3$

$(c)5$

$(d)2$

I am aware of the correspondence theorem for rings. Following that idea, the maximal ideals in $\mathbb{Z}_5[x]$ containing $\langle (x+1)^2(x+2)^3 \rangle $ are $\langle x+1 \rangle$ and $\langle x+2 \rangle$ which should give two maximal ideals in the given quotient ring. Am I right? Will the field $\mathbb{Z}_5$ cause some trouble here?

Please help.Thanks for your time.

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Am I right? Will the field $ℤ_5$ cause some trouble here?

Respectively: yes, no.

You have come to the correct conclusion. $ℤ_5$ helps you because then you are assured $ℤ_5[x]$ is a principal ideal domain.

Therefore the maximal ideals containing $(x+1)^2(x+2)^3$ correspond to the irreducible elements of its factorization, and you spotted those immediately.

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