Here is the final exam question in the last year semester that I've not solved.
Find the order of the $\mathbb{Z}_{10} [x] / \langle 5x^2 + 2x + 1 \rangle$
How Could I solve it?
Intuitive solution(my attempt)
Since $\mathbb{Z_{10}}[x] \simeq \mathbb{Z_2}[x] \times \mathbb{Z_5}[x]$
Then, $\mathbb{Z_{10}}[x] /\langle 5x^2 + 2x + 1 \rangle \simeq (\mathbb{Z_2}[x] / \langle 5x^2 + 2x + 1 \rangle) \times (\mathbb{Z_5}[x] / \langle 5x^2 + 2x + 1 \rangle$)
$\simeq $ $(\mathbb{Z_2}[x] / \langle x^2 + 1 \rangle) \times (\mathbb{Z_5}[x] / \langle 2x + 1 \rangle$)
Hence the order would be $20$
Sadly It might be false solution based on the my question" Is this statement true?(product and quotient of the groups and rings)"
Based on the above link, It looks like my solution is incorrect and wrong.
What is correct and fast solving method?
Any help would be appreciated.
The answer I remembered) $20$
Your solution is correct. In general, if $A=A_1\times A_2$ is a ring and $I\subset A$ is an ideal, then $I=I_1\times I_2$ with $I_i$ ideal of $A_i$, and $A/I\simeq A_1/I_1\times A_2/I_2$.
The difference with the link you provide is that it is not true that $A/I\simeq A_1/I\times A_2/I$ (mostly because in general it does not make sense).