How to characterize the maximal ideals of $\mathbb R[x]$ and $\mathbb C[x]$ ?

Question

It is known that if $M$ is a maximal ideal of $C[0,1]$ then for some $r \in [0,1]$ , $M=\{f \in [0,1] : f(r)=0\}$ , can we also characterize the maximal ideals of $\mathbb R[x]$ ? I think I somewhere read that there is a bijective correspondence between the maximal ideals of $\mathbb R[x]$ and points on the upper half plane but am not able to prove it . And what about maximal ideals of $\mathbb C[x]$ ?

Answer 1

For $\mathbb C[x]:$ First note that it's a PID, so every ideal is generated by a single polynomial. Also $\mathbb C$ is algebraically closed. So every non-constant polynomial splits into linear factors. Hence the maximal ideals of $\mathbb C[x]$ are of the form $\langle x-a\rangle,$ for some $a \in \mathbb C.$ In other words the maximal ideals of $\mathbb C[x]$ are in one-one correspondence with the points of complex plane.

For $\mathbb R[x]:$ The maximal ideals of $\mathbb R[x]$ corresponds to the irreducible polynomials over $\mathbb R.$ So the only possible choices are the linear factors and the irreducible polynomials of degree 2.

Added: Choose a degree two irreducible polynomial $f(x)$ over $\mathbb R$ and factorize it over $\mathbb C.$ Then we will have $f(x) = (x-\alpha)(x-\bar{\alpha})$ and at most one of the $\alpha, \bar{\alpha}$ will be in the upper half plane. Similarly, choose an element $\alpha$ in the upper half plane which is not real. Then $f(x) = (x-\alpha)(x-\bar{\alpha})$ will be a degree two irreducible polynomial over $\mathbb R.$ The linear polynomials corresponds to the point in the real line. This will give you the required correspondence between upper half plane and maximal ideals of $\mathbb R[x].$