A criterion for infinite non-Dickson nearfields of dimension two (Q910433)

Infinite nearfields which are not Dickson nearfields were constructed for the first time by H. Zassenhaus in 1987. However, there is a gap in the proof of the assertion that some of these nearfields are non-Dickson nearfields. In this paper a theorem is proved which states that if a nearfield N contains a field k in its center and the following three properties are satisfied: (i) N has dimension 2 over k, (ii) the factor group \(N^*/k^*\) contains no element of order 2, (iii) the characteristic of N is distinct from 2, then N is not a Dickson nearfield. This criterion is applied to some of the nearfields constructed by Zassenhaus, thus proving that these infinite nearfields are indeed non- Dickson nearfields.