Is ideal $(X^2+2X-3)$ prime in $\mathbb{Z}[X]$?

Question

I have this ideal in $\mathbb{Z}[X]$: $I=\left \langle X^2+2X-3 \right \rangle$.

Let's say $P$ and $Q$ are polynomials: $P(X)=a_0+a_1X+a_2X^2+...$, $Q(X)=b_0+b_1X+b_2X^2+...$.

$PQ=X^2+2X-3$. Since $PQ=a_0b_0+(a_1b_0+a_0b_1)X+...$, we have system:

$$-3=a_0b_0$$

$$2=a_1b_0+a_0b_1$$

and so on.

I'm not really sure what should be my next step. Can anyone help me?

Answer 1

Rational roots theorem:

$X^2+2X-3=(X-1)(X+3)$, so it is not irreducible, hence not prime.