I am a very basic field theory question. I must be mixing up a theorem here, but I am unsure which.
My goal here is to determine if there exists an inseparable, irreducible polynomial in a finite field.
Milne's book Fields and Galois Theory states that a nonconstant, irreducible polynomial $f \in F \left[ x \right]$ has a multiple root if $F$ has nonzero characteristic $p$, and $f$ may be written exclusively in the variable $x^p$.
More clearly, this second condition means there is some $g \in F \left[ x \right]$ so $f \left( x \right) = g \left( x^p \right)$.
This leads me to believe that $ f \left( x \right) = x^4 + x^2 +1$ is irreducible over the finite field of order 2, $F_2$.
Clearly it is nonconstant. $f \left( 0 \right) = f \left( 1 \right) = 1$, so it is irreducible over $F_2$. Lastly, if we define $g \left( x \right) = x^2 + x + 1 \in F_2 \left[ x \right]$, then $f\left( x \right) = g \left( x^2 \right)$. As the characteristic of $F_2$ is $2$, this seems to suggest the theorem holds, and so $f$ is inseparable over $F_2$.
However, later on Milne defines a perfect field as one over which all irreducible polynomial are separable. He says that a field with characteristic $p$ is perfect is every element of the field is a $p$-th power.
Now in $F_2$:
$0 = 0^2$ and $1^2 = 1$, so it seems like every element is a $2$-nd power, so $F_2$ should be perfect, and hence every irreducible polynomial is separable.
This two results seem to clash. Maybe the theorems in the book are poorly worded, or maybe I just keep misreading them.
Could someone please clarify what I have got wrong here? Also any comment about my overall goal, about whether there exist inseparable, irreducible polynomials over a finite field, would be appreciated too.
Best Answer
Indeed, $x^4+x^2+1$ is not irreducible over $\Bbb F_2$, since $$ x^4+x^2+1=(x^2 + x + 1)^2. $$