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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.

But I have no idea about others. Can someone please help? Thanks.

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Both 1 and 3 are correct.

  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.
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  • $\begingroup$ please explain (2) briefly,i can't understand $\endgroup$ – user274880 Sep 28 '15 at 15:00

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