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Based on this question I wonder:

(1) Is it possible, instead of taking just one variable $t$ and get $P:=\bigcup_{n\in \mathbb{N}}K((t^\frac1n))$, to take $m \geq 2$ variables $t_1,\ldots,t_m$ and get the ''$m$-Puiseux field", denote it by $P_m$.

(2) If (1) has a positive answer, is it true that if $K$ is an algebraically closed field of characteristic zero, then $P_m$ is also an algebraically closed field?

Remarks: Let $K$ be an algebraically closed field of characteristic zero.

(a) It was mentioned here that $P$ is isomorphic to the algebraic closure of $K((t))$ (= the field of formal power series over $K$), then perhaps $P_m$ is isomorphic to the algebraic closure of $K((t_1,\ldots,t_m))$, so $P_m$ is algebraically closed.

(b) Another way to show that $P_m$ is algebraically closed is by induction on $m$: When $m=1$ we know that $P$ is algebraically closed (by the above mentioned question). When $m=2$ we can view $P_2$ as the usual Puiseux field (in one variable $t_2$) over the algebraically closed field $\bigcup_{n\in \mathbb{N}}K((t_1^\frac1n))$, hence $P_2$ is algebraically closed (by the above mentioned question), etc.

  • I really apologize if my questions and remarks are trivial (either have a known answer or are just non-sense); it is because I am not so familiar with Puiseux fields.
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It seems that this article answers my question in the positive. It proves that $P_m$ is an algebraic closure of $K((t_1,\ldots,t_m))$.

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