  In 1907, Hans Hahn proved the remarkable fact that any ordered group can be
embedded in an ordered real function space. This set the stage for work on
ordered groups and fields, and this area received valuable contributions from
Levi-Civita and many others. In this paper we show how Hahn's method of
generating ordered fields from a base field and an ordered group actually
characterizes all complete ordered fields. We develop refinements of Hahn's
techniques, show that they are functorial in nature, and develop a new
invariant of an ordered field to prove that the decomposition of a complete
field into a base field and a "group of exponents" can itself further
decompose, uniquely, into a field combined with an arbitrary number of groups,
all of which are ordered and Archimedean.
