Prove the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series.
$F[[x]]$ is contained in $F((x))$. So there's at least a ring homomorphism that is injective. Can also see it's injective because the kernel of such a mapping would be trivial because $0$ is the same in either. Not sure if showing they are isomorphic is the best way to do this.
$\displaystyle \sum_{n \ge N} a_nx^n \in F((x))$
Im not sure how I would define the mapping. maybe theres a better way
The field of fractions of an integral domain is the smallest field that the domain injects into. The homomorphism that sends a power series to itself is an injective homomorphism into $F((x))$, since every power series is also a Laurent series. If $F$ is the field of fractions of $F[[x]]$, then $f$ injects into $F((x))$, so we just need to check that the smallest subfield of $F((x))$ containing $F[[x]]$ is $F((x))$ itself. But this is clear: since $x^n$ is a power series for all $n\geq 0$, then any subfield containing all power series contains $x^{-n}$ as well. Thus, such a subfield contains all sums of power series and finitely many negative powers of $x$, which is exactly the field of Laurent series $F((x))$. Thus, $F((x))$ is indeed the field of fractions of $F[[x]]$.
Probably the best way to think about it concretely is that the invertible elements of $F[[t]]$ are precisely the power series with non-zero constant term. This is obviously a necessary condition. Proving sufficiency is a good entry-level exercise. Hence any non-zero $f \in F[[t]]$ may be factored uniquely as $f = t^n g$ for $n \in \mathbb{Z}_{\geq 0}$ and $g(0) \neq 0$, and the units are exactly the case when $n=0$. In the field of fractions of $F[[t]]$, $t^n$ for $n>0$ will be invertible too, in which case we'll have $f^{-1} = t^{-n} g^{-1}$. Expanding $g^{-1}$ as a formal power series, $t^{-n} g^{-1}$ is precisely a formal Laurent series. Hence the field of fractions contains $F((t))$. This argument also makes it obvious that $F((t))$ is itself a field, so it must be the fraction field on the nose.
