Is the set of all polynomials whose coefficients are odd a prime ideal in $\mathbb{Z}[x]?$

Question

$1.$ The set of all polynomials whose coefficients are all even is a prime ideal in $\mathbb{Z}[x]$

This statement is true: take homomorphism from $\mathbb{Z}[x]$ onto $\mathbb{Z_2}[x]$.

Now my question is that

$2.$ Is the set of all polynomials whose coefficients are all odd is a prime ideal in $\mathbb{Z}[x]?$

Same logic in $1.$ or some other logic is there ?

Answer 1

For one thing, it is clear that multiplication by $2$ does not result in another such polynomial. Thus one of the conditions for an ideal isn't satisfied.