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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?

Jonny
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  • $\mathbf F_3((X))$ is the field of fractions of the ring of formal power series $$\mathbf F_3[[X]]$$ with coefficients in $\mathbf F_3$ – in other words the formal Laurent series with coefficients in $\mathbf F_3$. – Bernard Apr 17 '19 at 18:53
  • Before studying ramifications you should first recall the notions of function fields, power series rings, number fields etc. – Dietrich Burde Apr 17 '19 at 19:00

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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.

Dietrich Burde
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  • Thank you. It says that the extension is obtained by adjoining a primitive (p^f -1)st root of unity to K. – Jonny Apr 17 '19 at 19:58