Show $f(x)$ is irreducible
Let $$f(x)=x^4+x^2+t \in \mathbb{F}_2 (t)[x]$$
let $y=t$ $$f=y+x^2(x^4+1)y^0 $$
let $p=x^2$ use Eisensteins criterion.
basing it out of an example from hungerford algebra(there is a pdf online) right second example after Eisentins criterion (another way would be to use rational root test)
I would need something $D ( F_2)$ to be UFD then $D[x,t]$ is a UFD by thm 6.14
This is a theorem from lecture
Show that there are no irreducible polynomials $f(z) \in F_q [x]$ s.t $$gcd(f,f')=0 $$
using that theorem and treating $t$ like a constant in $F_2[x]$ $$ f'(x)=4x^3+2x=0$$
that would make $gcd(f,f')=gcd(f,0)=0$ so it is not irreducible which i think it means reducible (But its irreducible Contradiction!)
It is correct that $f$ is irreducible and it is also correct that there exists no irreducible $g \in \Bbb F_p[x]$ such that $\gcd(g,g')=0$.
The way to solve this contradiction is that $f$ is not an element of $\Bbb F_2[x]$, $f$ is an element of $\Bbb F_2(t)[x]$. You can't just "treat $t$ like a constant" and apply theorems about $\Bbb F_2$ to $\Bbb F_2(t)$, these are two different fields with quite different properties. (One has infinitely many elements and the other one has finitely many elements, for example).