On couplings on a simple transcendental extension (Q1430553)

Let \(F\) be a field and \(F^*=F\setminus\{0\}\). A function \(\kappa: F^*\to\Aut(F), x\to\kappa_x\) is called a coupling on \(F\) if \(\kappa_{x \kappa_x(y)}=\kappa_x \circ\kappa_y\) for all \(x,x\in F^*\). A coupling is called strong if \(\kappa_{\kappa_x(y)}=\kappa_y\) for all \(x,y\in F^*\). Let \(\kappa\) be a coupling on \(F=(F,+,\bullet)\). Let \(F=K(t)\), where \(t\) is transcendental over the field \(K\). Define \(\circ\) on \(F\) by \(x\circ y:=0\) if \(x=0\) and \(x\circ y=x\bullet \kappa_x(y)\) otherwise. Then \(F^\kappa:=(F,+,\circ)\) is a Dickson nearfield. Wahling determined the strong couplings \(\kappa\) on \(F=K(t)\) with \(\kappa(F^*) \subseteq \Aut_K(F)\). In this paper the author determines a more general class of couplings on \(F\)-the strong \((K\bullet t)\)-couplings. These couplings on \(F\) are a kind of a product of couplings \(\varepsilon\) with \(\varepsilon (F^*)\subseteq \Aut_K(F)\) and \(\varphi\) with \(\varphi(F^*)\subseteq \Aut_t(F)\). The author shows that there are essentially only three types of couplings \(\kappa\) with \(\kappa (F^*)\subseteq\Aut_K(F)\). A constructive description for these couplings is given. The class of couplings \(\kappa\) with \(\kappa(F^*)\subseteq\Aut_t(F)\) is determined and it is shown how such couplings can be constructed. Conditions are derived for when a product between a coupling \(\varepsilon\) with \(\varepsilon (F^*)\subseteq\Aut_K(F)\) and \(\varphi\) with \(\varphi(F^*)\subseteq \Aut_t(F)\) is defined. Such a product is called a \((K\bullet t)\)-coupling. If \(\kappa\) is a \((K\bullet t)\)-coupling, the characteristic of \(K\) is not 2, and the image of \(\kappa\) is not the Klein 4-group, then \(F^\kappa\cong F^{\kappa'}\), where \(\kappa'\) has a special form. There are four different possibilities for \(\kappa'\). All the possibilities for \(\kappa'\) are determined and it is shown how all such couplings \(\kappa'\) can be constructed. The results in this paper lead to new examples of nearfields.