# Extension of valuation in iterated Laurent series fields

according to well known theorems, every valuation of a field $$K$$ can be extended to a valuation of a field extension $$L$$ of $$K$$ and this can be explicit in the case of finite extensions.

My question is how explicit this can be made in the case of an iterated Laurent series field: If we have $$K=F((t_1))$$ for a field $$F$$ and $$L=K((t_2))=F((t_1))((t_2))$$, how can the extensions of the $$t_1$$-adic valuation on $$L$$ look like? Is there something that can be said about the valuation on $$L$$ (archimedean? complete? uniformizer...)?

Thanks!

• The $t_1$ valuation on $F((t_1))$ extends in an obvious way to $E((t_1))$ for any field extension $E/F$ such that every $a \in F((t_1)), a \not \in F$ is transcendental over $E$, so the goal is to find (using a transcendental basis of $F((t_1))((t_2))/F$ ?) such a $E$ such that $F((t_1))((t_2))\cong E((t_1))$, and the difficulty is that $E$ is not $F((t_2))$ – reuns Jan 23 at 16:28