0
$\begingroup$

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 $f(c)=(a,b)??$

  • 2
    dear @zorutic, I saw you got several previous questions anwsered but didn't accept any. You should consider to accept some of the answers (by clicking on the green mark) which were useful for you, so questions appears as answered :) – 2017-02-17

1 Answers 1

1

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.