some notation in field extension

Question

I know $\mathbb{F}_3$ = $\mathbb{Z}$/3$\mathbb{Z}$ = {0,1,2}. But what does $\mathbb{F}_3$ ((X)) mean? And how can we find a totally tamely ramified extension and unramified extension of it respectively?

Answer 1

The ring of formal Laurent series in one variable over a ring $R$ is denoted by $R((x))$, see here. In your case, $R=\Bbb F_3$. For the relation with the fraction field of $R[[x]]$ see here:

What is the fraction field of $R[[x]]$, the power series over some integral domain?

Concerning tamely ramified and unramified extensions see here.