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.


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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.