Planar closedness of Dickinson nearfields and the depth of group couplings (Q1187130)

A nearfield is called planar closed if all of its subnearfields are planar. In this paper two concepts related to planar closedness of nearfields are introduced, namely the depth of a coupling of a Dickson nearfield and the concept of term solutions of the planar equation, where a term is a term in the nearfield language. It is shown that if a nearfield has a finite number of term solutions, then it has a universal term solution. It can then be shown that the theory of a class of near fields is planar closed, if and only if the class has a universal term solution. Finally it is shown that a Dickson nearfield \(K\) coupled to a commutative field by a coupling of finite depth is planar closed if the Dickson group \(\Gamma_ K\) is periodic. Further, it is shown that the theory of \(K\) is planar closed if \(\Gamma_ K\) is of finite exponent. ``Bemerkung 1.1'' which is a statement about true couplings on arbitrary groups appears to be valid only for finite abelian groups.