Find a maximal ideal $I$ of $\mathbb{R} [x]$ such that $\mathbb{R} [x] / I$ not isomorphic as a ring to $\mathbb{R}$

Question

I want to find a maximal ideal $I$ of $\mathbb{R} [x]$ such that $\mathbb{R} [x] / I$ not isomorphic as a ring to $\mathbb{R}$. I can see that the maximal ideals of $\mathbb{R} [x]$ are ideals of the form $p \mathbb{R} [x] $ where $p$ is an irreducible polynomial, and I understand that the quotient of $\mathbb{R} [x]$ by a maximal ideal is a field. I am not sure how to show that it is not ismorphic to $\mathbb{R}$ though.

Answer 1

If $F$ is a field, then the maximal ideals of $F[x]$ are exactly the principal ideals generated by irreducible polynomials.

If $f$ is irreducible of degree $1$, then $F[x]/(f) \cong F$.

If $f$ is irreducible of degree $n > 1$, then $F[x]/(f) \not\cong F$ because $F[x]/(f)$ contains a root of $f$ but $F$ doesn't.

When $F=\mathbb R$, the irreducible polynomials are those of degree $1$ and those of degree $2$ with negative discriminant.