Determine whether $F^\times$ has a cyclic subgroup of order $10$ where $F=\mathbb{R}((t)),\mathbb{C}((t))$

Question

I wanted to verify my attempt:

Determine whether $F^\times=\langle F\setminus \{0\},\cdot, 1\rangle$ has a cyclic subgroup of order $10$ where:

1. $F=\mathbb{C}((t))$.
2. $F=\mathbb{R}((t))$.

1. We notice that $\langle \zeta\rangle\subset\mathbb{C}\subset F$ where $\zeta=\exp({\pi i\over5})$ is a cyclic subgroup of $F$ of order $10$.
2. If $\langle\alpha\rangle$ is a cyclic subgroup of $F$ of order $10$ then it must have that $\alpha\in\mathbb{R}((t))$ is a primitive unit root of order $10$. Then we must have $$\alpha=\exp({\pi i k\over 5})$$ where $k=1,2,3,4,6,7,8,9$. But in all of these cases $\alpha\notin\mathbb{R}((t))$. Thus we conclude that there isn't a cyclic subgroup of $\mathbb{R}((t))$ of order $10$.