# A problem on maximal ideal in polynomial ring.

Let $\Bbb R[x]$ be the polynomial ring over $\Bbb R$ in one variable. Let $I\subseteq\Bbb R[x]$ be an ideal. Then which are true?

1. $I$ is a maximal ideal if and only if $I$ is a non-zero prime ideal.
2. $I$ is a maximal ideal if and only if the quotient ring $\Bbb R[x]/I$ is isomorphic to $\Bbb R$.
3. $I$ is a maximal ideal if and only if $I=(f(x))$, where $f(x)$ is a non constant irreducible polynomial over $\Bbb R$.
4. $I$ is a maximal ideal iff there exists a non constant polynomial $f(x)\in I$ of degree $\le2$.

I know $\Bbb R$ is a field implies $\Bbb R[x]$ is Euclidean domain implies principal ideal domain, so (1) is true. I think in PID (3) is also true.

1. Consider the polynomial $X^2+1 \in \mathbb{R}[X]$. Can you find out the quotient by its ideal (which in this case is maximal) ?
Answer - $\mathbb{R}[X]/(X^2+1) \simeq \mathbb{C}$ via the mapping $\bar{X} \to i$. Can you now work out the details?
1. Can there exist an irreducible polynomial of degree $\geq 3$ in $\mathbb{R}[X]$? Fundamental Theorem Of Algebra helps.