By their definition, valuations are homomorphisms from the multiplicative
group of a field into ordered abelian groups (subject to the ultrametric inequal-
ity). It is therefore not surprising that many results about valuations reduce to
general facts about the category of ordered abelian groups, or even that of torsion-
free abelian groups. Of course, the latter categories have a simpler structure than
that of valued fields, and therefore allow simpler proofs. For this reason we de-
velop in this part of the book several themes from the theory of abelian groups
which will ultimately be applied in field-theoretic considerations, but do so in a
purely group-theoretic manner. For the general theory of abelian groups see [Fu]
or [Kap2].
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