nLab
Hahn series

Definition

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

(fg)(x)= y+z=xGf(y)g(z)(f \cdot g)(x) = \sum_{y+z = x \in G} f(y)g(z)

The multiplicative valuation v(f)v(f) is the least xG opx \in G^{op} for which f(x)0f(x) \neq 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]k[x^G] is a field. If kk is algebraically closed, then k[x G]k[x^G] is algebraically closed provided that GG is divisible.

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

Revised on February 5, 2012 03:27:18 by Todd Trimble (74.88.146.52)