TITLE: Maximal ideals in polynomial rings with real and complex coefficients
QUESTION [5 upvotes]: I was asked in homework to think about maximal ideals in polynomial rings $\mathbb{R}[x]$ and $\mathbb{C}[x]$. I have realized that: $\forall c\in\mathbb{R},\;I_c : = \{p(x)\in\mathbb{R}[x]\;|\;p(c) = 0\}$ is an ideal (similar for $\mathbb{C}[x]$), now in order to prove it to be maximal, I need to show:
$$I_c\subset J\subsetneq A,\;J\text{ is an ideal}\Longrightarrow I_c = J$$
which I have difficulty showing.
Secondly, I don't know how to show that all maximal ideas are in the form of $I_c$. Some help please. Thank you.

REPLY [5 votes]: Hint $\ $ Polynomial rings over fields enjoy a (Euclidean) division algorithm, hence every ideal is principal, generated by an element of minimal degree (= gcd of all elements). But for principal ideals: contains $\!\iff\!$ divides, i.e. $\rm\: (a)\supseteq (b)\!\iff\! a\mid b.\:$ Thus, having no proper containing ideal (maximal) is equivalent to having no proper divisor (irreducible).  Summing up, we have deduced that, in a PID, $\rm\ (f)\:$ is maximal $\rm\!\iff\! f\:$ is irreducible. Thus the problem reduces to determining the irreducible polynomials in $\rm\,\Bbb R[x]\,$ and $\rm\,\Bbb C[x],\,$ which is straightforward.