Proving $\mathbb C[x,y]/\langle x^2+y^2+1\rangle,\mathbb R[x,y]/\langle x^2+y^2+1\rangle$ are integral domains

Question

As a homework assignment,

I need to prove that $\mathbb C[x,y]/\langle x^2+y^2+1\rangle$ and $\mathbb R[x,y]/\langle x^2+y^2+1\rangle$ are integral domains.

I have no idea how to approach problems like this. We're allowed to use the fact that for any field, $\mathbb F[x,y]/ \left\langle xy+b \right\rangle $ is an integral domain iff $b\neq 0$.

Here are some thoughts: If $x^2+y^2+1$ is irreducible then it generates a prime ideals and we're done. Otherwise, since each of the polynomials rings is a UFD, we may factor it into irreducibles. Then I thought of using the chinese remainder theorem somehow to simplify into the information I was given, but I don't see how. How should I solve these problems?

Answer 1

For $\mathbb C$, you can use the fact you stated by a change of variables: observe that $x^2+y^2+1 = (x+iy)(x-iy)+1$, and rewrite $\mathbb C[x,y]$ as $\mathbb C[w, z]$ with $w=x+iy$ and $z=x-iy$.

For $\mathbb R$, my guess is you are supposed to use the $\mathbb C$ case; show that the kernel of the homomorphism $\mathbb R[x,y] \to \mathbb C[x,y]/\langle x^2+y^2+1\rangle$ given by $x \mapsto x$ and $y \mapsto y$ is precisely the ideal in $\mathbb R[x,y]$ generated by $x^2+y^2+1$. This will induce an embedding of $\mathbb R[x,y]/\langle x^2+y^2+1\rangle$ into $\mathbb C[x,y]/\langle x^2+y^2+1\rangle$, witnessing that the former is an integral domain since the latter is.

Answer 2

For any field $K$ of characteristic not equal to $2$, the polynomial $x^2+y^2+1$ is irreducible in $K[x,y]$ (a sketch of a proof of this claim is provided in the hidden section below, but the idea is to use Eisenstein's Criterion with $K[y][x]=K[x,y]$). As $K[x,y]$ is a unique factorization domain, $x^2+y^2+1$ is also prime in $K[x,y]$, whence the ideal $\left(x^2+y^2+1\right)$ is a prime ideal. Therefore, $K[x,y]/\left(x^2+y^2+1\right)$ is an integral domain.

It remains to show that $x^2+y^2+1$ is irreducible over $K$. This can be done by applying Eisenstein's Criterion to $x^2+y^2+1$, treating it as an element of $K[y][x]$. Show that there is no irreducible element $p(y)\in K[y]$ such that $\left(p(y)\right)^2$ divides $y^2+1$ (this will be the reason why $K$ can't be of characteristic $2$).

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If $K$ is of characteristic $2$, then $x^2+y^2+1=(x+y+1)^2$, so $K[x,y]/\left(x^2+y^2+1\right)$ has a nonzero nilpotent element, namely the image of $x+y+1$ under the quotient map $K[x,y]\to K[x,y]/\left(x^2+y^2+1\right)$.

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Question: Prove that, if $K$ is a field of characteristic not equal to $3$, then $K[x,y]/\left(x^3+y^3+1\right)$ is an integral domain.

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Edit: G. Sassatelli had already mentioned in the comment above what I did.