Let $I=\langle 5,X^2+Y+1\rangle\subset R=\mathbb Z[X,Y]$. Show that $R/I\cong\mathbb F_5[X]$.

Question

Let $I=\langle 5,X^2+Y+1\rangle\subset R=\mathbb Z[X,Y]$. Show that $R/I\cong\mathbb F_5[X]$.

My book suggests to use four theorems, which are in short ($J\subset I$ are ideals):

1) the isomorphism theorem $R/I\cong(R/J)/(I/J)$,

2) $R[X]/I[X]\cong (R/I)[X]$,

3) $R[X]/\langle X-\alpha\rangle\cong$ R,

4) $\langle a,b\rangle=\langle a,b+ca\rangle$.

Now I'm guessing I could use 4) to write $I$ in a different way, but I wouldn't know how. Also, I'm opting for $J=\langle 5\rangle$. So we would get $$ R/I=(R/\langle 5\rangle)/\langle X^2+Y+1+\langle 5\rangle\rangle. $$

I'm not really show how to use 2) and 3) though.. and I'm just clueless overall. Could someone give me a hint?

Answer 1

Hint: according to (3), what is $(\mathbb{Z}[X])[Y]/\langle Y + X^2+1 \rangle$?