Consider the evaluation map $E_2: \mathbb{F}_2[x]\mapsto \mathbb{Z},\quad E_2(f(x)) = f(2)$
Do all irreducible polynomials in $\mathbb{F}_2[x]$ map to a prime number when using that map?
For instance, $x^2+x+1$ is irreducible. And if you use the above map, we have $2^2+2+1 = 7$. This is also true for $E_2(x^3+x^2+1) = 13$, $E_2(x^5+x^2+1) = 37$, $E_2(x^8+x^4+x^3+x+1) = 283$
And as we can see all those numbers are prime. Hence, if $p(x) \in \mathbb{F}_2[x]$ is irreducible, does that implies that $p(2) \in \mathbb{Z}$ is prime?
No. The polynomial $$ f=x^4+x^3+1 $$ is irreducible over $\Bbb F_2$, but $f(2)=25$ is not a prime number.