I’m studying abstract algebra for my finals and I have a question. I’m asked to show that the quotientring underneath is a field.
$$ \frac{\mathbb{Z} \left[ X, Y \right]}{(5, XY-Y+6, Y-X^2)}$$
Due to the correspondence theorem we can quotient out sequentially, such that we get:
$$ \frac{\mathbb{Z}_5 \left[ X, Y \right]}{(XY+4Y+1, Y-X^2)}$$
Where the ideal now also lies in $\mathbb{Z}_5 \left[ X, Y \right]$. Then I quotient out by $Y-X^2$, such that the ring is isomorphic (via the first isomorphism theorem) to:
$$ \frac{\mathbb{Z}_5 \left[ X, X^2 \right]}{(X^3+4X^2+1)} \cong \frac{\mathbb{Z}_5 \left[ X \right]}{(X^3+4X^2+1)} $$
Now $\mathbb{Z}_5[X]$ is a PID and the ideal between brackets is not maximal, because the polynomial is reducible in $\mathbb{Z}_5$, so the quotientring can't be a field. Where have I gone wrong in my approach, if it is completely faulty, how should I approach these type of questions?
Thanks in advance