If we define trace to be $x+x^p+\cdots+x^{p^{n-1}}$. How do we know there is an element of nonzero trace? Clearly if $a\in F_p$ then its trace is zero as $a^{p^i}=a$ so $\operatorname{tr}(a)=a+a+\cdots+a=pa=0$. So we know this element has to come from $F_{p^n}$ and I reckon it needs to not be in any subfield, but I can't seem to show this.
Also, I am trying to show that $x^p-x-a$ which is in $F_{p^n}$ is either irreducible or factors completely. So far, I have seen that if the polynomial has some root, say $b$, then it also has roots $b+i$ for $i=1$ to $p-1$ so we found all the $p$ roots of the $p$ degree polynomial so it factors completely into linear factors if we can get one of its roots. But I don't see why it's that or irreducible. We can't the polynomial factor into two polynomials of lower degree that are irreducible?