Testing for irreducibility over $R=\mathbb Q[x]/(1+x^2)$

Question

Let $R=\mathbb Q[x]/I$ where $I$ is the ideal generated by $1+x^2$. Then is

1. $y^2 +1$ is irreducible over $R$ ?
2. $y^2+y+1$ is irreducible over $R$ ?
3. $y^2-y+1$ is irreducible over $R$ ?
4. $y^3+y^2+y+1$ is irreducible over $R$ ?

I am completely stuck on it. Please help.

Answer 1

You can basically think of this quotient ring as "$\mathbb{Q}$ with $\sqrt{-1}=x$ adjoined", or "the complex numbers with rational coefficients". (Fun to say.)

That said, #1 obviously has $x$ as a root.

For 2 and 3, you could check to see what their real roots look like with the quadratic formula... if those roots are in $R$, then they're reducible.

For 4 at the very worst you could actually substitute $ax+b$ and solve the resulting system of equations to see if you can get a solution in your extension. Or you can apply the rational root test and see if it has any rational roots immediately.

I get "no, yes, yes, no"