Maximal and Prime Ideals I was assigned these problems for homework to designate if they were maximal, prime or neither.  I was able to determine that (a) was solely prime by showing $\mathbb{Z}[x] /(x-1)$ is isomorphic to $\mathbb{Z}$ which is an integral domain.  I am having troubles in answering the other 3 though and could use some help. Thanks.
(a) $(x − 1)$ in $\mathbb{Z}[x]$
(b) $(x − 1)$ in $\mathbb{R}[x]$
(c) $(x^2 + 1)$ in $\mathbb{R}[x]$
(d) $(x^2 + 1)$ in $\mathbb{C}[x]$

Hints: 
(b) See if you can show that $\mathbb{R}[x]/(x-1) \cong \mathbb{R}$ in a similar manner to (a). From this, you conclude...
(c) Is $(x^{2}+1)$ reducible over $\mathbb{R}[x]$? What do you know about irreducible elements of PIDs?
(d) Is $(x^{2}+1)$ reducible over $\mathbb{C}[x]$?
