The sum of all primitive $n$-th roots of unity in the algebraic closure of $F(x)$

Question

If $n$ is any natural number, then $\mu(n)$ is the sum of all primitive $n$-th roots of unity in $\mathbb{C}$ where $\mu$ is the Möbius function defined as $\mu(n)=0$ if $n$ is not square free and $\mu(n)=(-1)^k$ otherwise where $k$ is the number of primes dividing $n$. Is it true in function fields? Say $F$ is a finite field and $F(x)$ is the field of fractions of the polynomial ring $F[x]$. Is it true that if take the algebraic closure of $F(x)$ and any polynomial $f$, the sum of $|f|$-th roots of unity is $\mu(f)$? Or is there an easy way to compute the sum of all primitive $n$-th roots uf unity in the algebraic closure of $F(x)$?