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?