A polynomial $f(x)=a_0+a_1x+......+a_nx^n\in R[x]$ where $R[x]$ is a polynomial ring over a ring $R$ is said to be primitive if $\gcd(a_0,a_1,a_2,......,a_n)$ is a unit.
I could find examples of polynomials which are reducible but not primitive,primitive but not reducible. I cant find an example of a non-primitive but irreducible polynomial .Any help
The answer depends on what you mean by irreducible. Is the polynomial $2x^2+2 \in \Bbb Z[x]$ irreducible?
If you say no because it factors as $2\cdot(x^2+1)$, then no nonprimitive polynomial can be irreducible: you can always factor it as $\gcd(a_0,\dots,a_n)$ times another polynomial.
If you say yes because the above factorization involves a degree-0 polynomial, then it is the example you seek.
