I'm currently learning for my Commutative Algebra exam and I have the following question:
Given $\mathbb Z$ the ring of all integers and the ring $\mathbb Z[X, Y]$ the ring of polynomials in indeterminates X and Y with coefficients in $\mathbb Z$. We also consider the factor ring:
$R = \mathbb Z[X, Y]/(Y^2+1)$,
I noticed that $5$ is not a prime in $R$, since we can decompose
$Y^2 + 1 = (Y-2)(Y-3)$ in $\mathbb Z_5$.
My question is:
How do I determine the decomposition of $5$ into prime factors in $R$ ? (I need this in order to compute the irredundant primary decomposition of an ideal in this given ring $R$).
More general, how can we determine the decomposition of an element in prime/irreducible factors in such rings of polynomials?
Thank you!
Observe that $R \cong \mathbb{Z}[i][x]$ and then use the usual tools (norm map $N \colon \mathbb{Z}[i] \rightarrow \mathbb{Z}$, the ring is an integral domain, so we can argue by degree of polynomials etc.) - one finds $N(5)=25$, so if there is a non-trivial decomposition $5=fg$, then $f$ and $g$ are elements of $\mathbb{Z}[i]$ with $N(f)=N(g)=5$. But $x^2+y^2=5$ has (upto sign permutation) only one solution - namely $x=2$, $y=1$ - which in our case provides a decomposition $5=(2+i)(2-i)$.
Transporting this decomposition via the isomorphism, we have in $R$ that $5=(2+Y)(2-Y)$.