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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!

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  • $\begingroup$ 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 $F((t_2))$ $\endgroup$ – reuns Jan 23 at 16:28

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