Puiseux series

Remarks

Example: $f = X^{1/2}$ is a Puiseux series with $m = 2$, and it is a root of $Y^2 - X$ in the field of Puiseux series, since $(X^{1/2})^2 = X$ in $k((X^{1/2}))$. Non-example: the formal sum $\sum_{n \ge 1} X^{1 - 1/n}$ has the exponents $0, 1/2, 2/3, 3/4, \ldots$, whose denominators are unbounded, so there is no single $m$ and it is not a Puiseux series. The union is a field because any two Puiseux series lie in a common field $k((X^{1/m}))$. The order $v(f)$ of the lowest exponent with nonzero coefficient is a valuation on the field of Puiseux series with value group $\Q$.