Show that the principal ideal

in

is prime but not maximal.

To show it is prime consider

. Then we want to show that

or

.

Now

. So

for some polynomial

. This implies that either

or

since we can take the other to be the unit polynomial.

Uuh?? What "other"? And what is "the unit polynomial"? Perhaps this is simpler: a polynomial is divisible by (x-1) iff 1 is one of its roots (this is the residue theorem for (long) division of polynomials), so
Suppose for contradiction that it was maximal. Then if

, either

or

. Consider

. Then it doesn't contain

. Thus

is not maximal.

Is this correct?