Hahn series

Remarks

For each $\gamma$ the sum defining $(fg)(\gamma)$ has finitely many nonzero terms, and the support of $fg$ is again well-ordered. With these operations $K((t^\Gamma))$ is a field, as Hahn showed. Example: for $\Gamma = \Z$, the well-ordered subsets are the subsets bounded below, so $K((t^\Z))$ is the field of formal Laurent series $K((t))$. Example: for $\Gamma = \Q$, the series $\sum_{n \ge 1} t^{1 - 1/n}$ is a Hahn series, since its support $\set{0, 1/2, 2/3, \dots}$ is well-ordered. Non-example: for $\Gamma = \Z$, the formal sum $\sum_{n \ge 1} t^{-n}$ is not a Hahn series, since its support $\set{-1, -2, \dots}$ has no least element. The map $v(f) = \min \set{\gamma : f(\gamma) \ne 0}$ for $f \ne 0$ is a valuation on $K((t^\Gamma))$ with value group $\Gamma$.