Cyclotomic extension of $\mathbb{F}_p((T))$

Question

I feel very confused about why adding n-th roots of unity to $\mathbb{F}_p((T))$ would give $\mathbb{F}_{p^n}((T))$. (Is this true?)

Answer 1

No, it is not true. Simplest counterexample? It’s $p=2$, $n=3$, where the cube roots of unity are quadratic over the base, just as they are over $\mathbb Q$. So the extension field is $\mathbb F_4((T))$,