nLab
Hahn series

Definition

The ring of Hahn series with value group? $G$, denoted $k[x^G]$, is the ring of functions $f:G\to k$ such that $\{x\in G:f(x)\ne 0\}$ is well-ordered when considered as a subset of the opposite poset $G^{\mathrm{op}}$. Addition is defined pointwise, and multiplication is defined by the convolution product:

$$(f\cdot g)(x)=\sum _{y+z=x\in G}f(y)g(z)$$(f \cdot g)(x) = \sum_{y+z = x \in G} f(y)g(z)

The multiplicative valuation $v(f)$ is the least $x\in G^{\mathrm{op}}$ for which $f(x)\ne 0$.

We obtain a valuation ring from this construction since a valuation ring determines and is determined by a valuation on a field.

Theorem

The ring $k[x^G]$ is a field. If $k$ is algebraically closed, then $k[x^G]$ is algebraically closed provided that $G$ is divisible.

As a corollary, if $G$ is divisible, $k[x^G]$ is real closed if $k$ is real closed. This is because the adjunction of a square root of $-1$ would make $k[x^G]$ algebraically closed, since this gives the same result as constructing the Hahn series over the algebraically closed field $k[\sqrt{-1}]$.