Embedding of near-fields with real valuations into planar near-fields
The main result of this note is: Let \((F,F^{\phi})\) be a Dickson pair of near-fields, where \(\phi\) is a coupling homomorphism. If there exists a real valuation \(V: F\to\Gamma \cup\{-\infty\},\) where, as usual, \(\Gamma (+,<)\) is an ordered subgroup of the reals to which a smallest element -\(\infty\) is added, then \(F^{\phi}\) can be completed relative to the topology induced by V, and the completion is planar, provided the following conditions are satisfied: (i) V x\({}_{\phi}=V\) for all \(x\in F\backslash\{0\}\) and (ii) \(\ker V\subset U_{\phi}.\) This extends results by \textit{H. Wefelscheid} [On the immersion of non-planar near- fields into planar near-fields, Near-rings and near-fields, Proc. Conf., San Benedetto del Tronto/Italy 1981, 219-224 (1982)].
scientific article; zbMATH DE number 3875425