# Show that ⟨2,x⟩\langle 2,x \rangle is not a principal ideal in Z[x]\mathbb Z [x]

Hi
I don’t know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please.

The ring that I am talking about is $\mathbb{Z}[x]$ so $\langle 2,x \rangle$ refers to $2g(x) + xf(x)$ where $g(x)$, $f(x)$ belongs to $\mathbb{Z}[x]$.

I think it’s relatively easy to see that $$I=⟨2,x⟩={anxn+⋯+a1x+a0;a0 is even}I=\langle 2,x \rangle = \{a_nx^n+\dots+a_1x+a_0; a_0\text{ is even}\}$$.
Now, suppose that $$I=⟨f(x)⟩I=\langle f(x) \rangle$$ for some $$f(x)∈If(x)\in I$$.
If $$f(x)f(x)$$ is a constant polynomial, then $$⟨f(x)⟩\langle f(x) \rangle$$ contains only polynomials with even coefficients, and we do not get $$xx$$.
If $$f(x)f(x)$$ is of degree at least $$11$$, then non-zero polynomials in $$⟨f(x)⟩\langle f(x) \rangle$$ have degree at least $$11$$, and we do not get $$22$$.
So $$II$$ is not of the form $$⟨f(x)⟩\langle f(x) \rangle$$.