It is easy to prove that for a an irreducible polynomial $P$ of degree $d$ of $\mathbb F_q[T]$, one can embed $\mathbb F_{q^d}$ in $\mathbb F_q(T)_P$ (the completion of $\mathbb F_q(T)$ at $P$) and there exists an element $\alpha\in\mathbb F_{q^d}$ such that $\mathbb F_q(T)_P=\left\{\sum_{n\ge -m} a_n(T-\alpha)^n\mid \alpha_n\in\mathbb F_{q^d},\,m\in\mathbb Z\right\}$
I am looking for a reference where this kind of things is proved.
Thanks in advance
Have you tried using Hensel's lemma to show $|T - \alpha|_P < 1$ for some $\alpha$ in $\mathbf F_{q^d}$ and then show $T-\alpha$ is a uniformizer in the completion?
Once you have a uniformizer $t$ in the completion, show $\mathbf F_{q^d}[t]$ is dense in the valuation ring and then that the valuation ring is $\mathbf F_{q^d}[[t]]$ with the $t$-adic topology, so the completion (the fraction field of the valuation ring) is the Laurent series field \mathbf F_{q^d}((t))$. Use $t = T-\alpha$ and then you're done.
