Orderings, cuts and formal power series

Abstract

Each real closed field 'R' can be viewed as a subfield of the formal power series field [kappa](('G')), where ' G' and [kappa] are respectively the value group and the residue field of the natural valuation 'v' on 'R'. One can prove that cuts in a given Hahn product 'H' might be realized by suitable elements of a certain bigger Hahn product. In this way, each cut in 'R', and hence each ordering on 'R'(' y') corresponds to a canonically defined element [psi] in a certain bigger formal power series field [kappa]α ((' G'α)) which contains [kappa](('G')). These elements [psi] do not belong to 'R'. More generally, one can prove that orderings on the ring 'R'['y'] are in one-to-one correspondence with canonically defined elements [straight phi], where [straight phi] is an element [psi] as above or an element of 'R'. To find [straight phi], one first fixes an embedding [iota] of 'R' into [kappa](('G')), which is also proper, i.e., the value group of [iota]('R') is 'G'. Then it can be seen that there exists a certain extension [kappa]α((' G'α)) of [kappa](('G')), so that [straight phi] ∈ [kappa] α(('G'α)). The correspondence between the orderings on the ring 'R'[' y'] and the elements [straight phi] can be generalized to the case of the orderings on the ring 'R'['y'1,···,' yn']. Actually, one can obtain an 'n'-tuple ([straight phi]1,···,[straight phi]'n') corresponding to an ordering on 'R'['y' 1,···,'yn'], where all the [straight phi]'i''s belong to a certain Hahn product. If 'F' is an ordered field having 'R' as its real closure such that [iota]('F') ⊆ [kappa]((' V')), where 'V' is the value group of 'F', then it can be proved that the value group 'W' of ' F'([straight phi]1,···,[straight phi]' n') is generated over 'V' by all the exponents γ appearing in some [straight phi]'i', 1