Showing (2) + (x) is not a principal ideal in Z[x]

Hi there, I am struggling to understand this question and am unconvinced with my approach. So a principal ideal is one where each element can be written as a 'multiple' of the other.

So (2) + (x) = { a2 + bx | a , b [x] }, it seems too simple to just plug in values for a and b to get two elements of the ideal and then show they are not multiples of each other.. Also I am unsure on the second part of the question as
is not a field(i think).

Well the ideal generated by 2 and x is equivalent to

The key thing to notice is that any polynomial in this ideal will have even constant term (because once you multiply g(x) by x, you no longer have a constant term and so the constant of any polynomial in the ideal comes from the 2f(x) which will clearly be even).

Now, suppose that this ideal is principal. So there exists some such that . You quite quickly get a contradiction to this, suppose h(x) were a constant polynomial - what's wrong with this? Then suppose it's a non-constant polynomial of degree at least one - you fail to include which elements of (2,x)?

is not a field, it's a ring (because inverses to elements do not belong to the set). To show is not a PID you want to use induction. You've already done the base step by showing (2,x) is not principal, you just need to adapt the argument slightly to do the inductive step.