I'm doing Exercise 1 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.
In $\mathbb Z[x]$, show that the set $A$ of all polynomials with even constant term is not a principal ideal.
Because my solution is so short that I suspect it contains subtle mistake. Could you please verify if my attempt is fine or contains logical mistakes? Thank you so much for your help!
The product of an even integer with any other integer is even. The sum of even integers is again even. Then $A$ is an ideal.
We have $p := x^2+x+2 \in A$ is irreducible over $\mathbb Z$. If $A$ is principal, then $A = \langle p\rangle$. However, $q=2 \in A$ is not divisible by $p$. Hence $A$ is not principle.