Embedding ordered fields in formal power series fields (Q1602685)

Let \(F\) be an ordered field. The aim of the paper is to investigate how an embedding of \(F\) into a formal power series field can be extended canonically to an embedding of a simple extension \(F(y)\) of \(F.\) Let \(v\) be the finest valuation on \(F\) compatible with the ordering, \(V\) the value group and \(k\) the residue group of \(v.\) If \(F'\) is an ordered extension of \(F\) then the finest valuation on \(F'\) compatible with the extended ordering is denoted also by \(v.\) The value group and the residue field of the extended \(v\) are denoted by \(V'\) and \(k'\), respectively. Denote the real closure of \(F\) by \(\overline F.\) Now the residue field is \(\overline k,\) the real closure of \(k\), the value group is \(\overline V\), the divisible hull of \(V.\) An embedding \(p:\overline F\rightarrow \overline k((\overline V))\) is called proper if \(v(p(z))=v(z)\) for all \(z\in \overline F.\) The following embedding theorem is proved: Let \(p:\overline F\rightarrow\overline k((\overline V))\) be a proper embedding, \(F'=F(y)\) be an ordered extension of \(F.\) Then there exists a canonically defined power series \(\phi\in \overline {k'} ((\overline {V'}))\) (depending on \(y\)) such that: 1) \(p\) extends to a proper embedding \(p:\overline {F'}\rightarrow\overline {k'} ((\overline {V'}))\) via \(y\mapsto\phi.\) 2) If \(p(\overline F)\) is truncation closed then so is \(p(\overline {F'}).\) 3) If \(p(F)\subseteq {\sim \atop k}((V)),\) then \(\phi\in {\sim \atop k'}((V')).\) Equivalently, \(p(F')\subseteq{\sim \atop k'}((V')),\) equivalently, \(V'=V_1\) and \(k={\sim \atop k_1}\) where \(V_1\) is the group generated by \(V\) and the support of \(\phi\) and \(k_1\) is the field generated by \(k\) and the coefficients of \(\phi\). It is proved that under some natural restrictions the extended value group \(V'\) can be prescribed arbitrarily. For uncountable \({\aleph}_{\alpha}\), divisible ordered abelian groups (respectively, real closed fields) are \({\aleph}_{\alpha}\)-saturated as ordered groups (respectively, real closed fields) iff they are \({\aleph}_{\alpha}\)-saturated as ordered sets, that is, iff they are \({\eta}_{\alpha}\)-sets [\textit{S. Kuhlmann}, Groupes abéliens divisibles ordonnés, Publ. Math. Univ. Paris VII 32, Séminaire sur les Structures Algébriques Ordonnées. Vol. 1, Selection d'exposés 1984-1987, 3-14 (1990; Zbl 0704.06010)]. In the present paper necessary and sufficient conditions are proved for a real closed field to be \({\aleph}_{\alpha}\)-saturated, where \({\alpha}\geq 0\).