Let $K$ be a field, and $L$ be an algebraic extension of $K$. I think it is known that if $T$ is a finite extension of $K((X))$, then $T$ is complete with respect to the $X$-adic valuation, hence if $T$ is included in $L((X))$, $T= N((X))$ for some finite extension $N$ of $K$ (true ?).
1) Is $L((X))$ an algebraic extension of $K((X))$ ?
2) assuming that 1) is true, am I right to say that if $L/K$ is Galois, then Gal($L/K$) is isomorphic to Gal($L((X))/K((X))$) ?
thx.