In How many elements there exist in polynomial quotient ring $\mathbb{Z}_5[X]/(X^2+1)?$ I asked the number of elements in this polynomial ring.
Now I'd like to ask its CRT.
Since $ (\mathbb{Z}_5[X]/X^2+1)\cong (\mathbb{Z}_5[X]/X+2)\times(\mathbb{Z}_5[X]/X+3) $,
We can view $(\mathbb{Z}_5[X]/X^2+1)$ as two-dimensional vector space over $\mathbb{Z}_5[X]$ where this two refers to the number of factors.
What is the mapping of element in $(\mathbb{Z}_5[X]/X^2+1)$ to that in $(\mathbb{Z}_5[X]/X+2)\times(\mathbb{Z}_5[X]/X+3)$ ??
In the previous question, the number of elements in $(\mathbb{Z}_5[X]/X^2+1)$in equivalent to the product $|(\mathbb{Z}_5[X]/X+2)|\cdot|(\mathbb{Z}_5[X]/X+3)|=5・5=25.$
So there should exist one-to-one mapping between this isomorphic ring.
The mapping means as follows:
$a\in \mathbb{Z}_5[X]/(X+3),\\ b\in \mathbb{Z}_5[X]/(X+2),\\ c\in \mathbb{Z}_5[X]/(X^2+1)$
$c \mapsto (a,b)$
Can anyone explain how to find the mapping of elements? More specifically, what is the actual formula of function $f$ of form $f(c)=(a,b)??$
According to the CRT, the mapping $\mathbb{Z}_5[X]/(X^2+1) \to \mathbb{Z}_5[X]/(X+2) \times \mathbb{Z}_5[X]/(X+3)$ that sends the class of $P$ mod $X^2+1$ to $(P \bmod(X+2), P \bmod (X+3))$ is well defined, and is a desired isomorphism.