2 minor edit

I think your argument is essentially correct.

Here is a proof for the algebraic closure of a finite field. It is enough to deal with the units of the ring of truncated Witt vectors $W_n(k)$ for all $n$. But then this is an algebraic group over a finite field and a theorem of Lang (Amer J Math 1956) states that $x \mapsto \sigma(x)x^{-1}$ is surjective for any algebraic group over such a field. I think from the algebraic closure of a finite field, the result follows for any algebraically closed field of positive characteristic.

It's not going to hold for any finite field, as you'll get the elements of norm one only. For $\mathbb{F}_p$ you don't get the identity but the function identically equal to $1$.

1

I think your argument is essentially correct.

Here is a proof for the algebraic closure of a finite field. It is enough to deal with the units of the ring of truncated Witt vectors $W_n(k)$ for all $n$. But then this is an algebraic group over a finite field and a theorem of Lang (Amer J Math 1956) states that $x \mapsto \sigma(x)x^{-1}$ is surjective for any algebraic group over such a field. I think from the algebraic closure of a finite field, the result follows for any algebraically closed field of positive characteristic.

It's not going for any finite field, as you'll get the elements of norm one. For $\mathbb{F}_p$ you don't get the identity but the function identically equal to $1$.