How many ideals contain $(x^2+1)$ in $\mathbb{Z}_4[x]$?

I presume $\Bbb Z_4$ means $\Bbb Z/4\Bbb Z$.

The answer to your question is "the number of ideals of the quotient ring $\Bbb Z_4[X]/(X^2+1)$". But $$R=\Bbb Z_4[X]/(X^2+1)\cong \Bbb Z[X]/(4,X^2+1)\cong\Bbb Z[i]/(4).$$ The ring $\Bbb Z[i]$ is a PID, and the ideals containing $(4)$ are generated by the factors of $4$. But $(4)=((1+i)^4)$ as ideals in $\Bbb Z[i]$, so these ideals are $((1+i)^k)$ for $0\le k\le 4$. There are five of them.

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Polynomials

Abstract Algebra

Ring Theory

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