On couplings on a simple transcendental extension

Abstract

H. Wähling [7] determined the strong couplings κ on F = K(t) with κ(F*) ⊂ Aut_K(F).¹ In this paper we determine a more general class of couplings on F — the strong (K · t)-couplings. These couplings on F are a kind of a product of couplings ε with ε(F*) ⊂ Aut_K(F) and φ with φ(F*) ⊂ Aut_t(F). In the first section we show that there are essentially only three types of couplings κ with κ(F*) ⊂ Aut_K(F), and we give a constructive description for these couplings. In the second section we determine the class of couplings κ with κ(F*) ⊂ Aut_t(F) and show how such couplings can be constructed. In the third section we derive conditions for when a product between a coupling ε with ε(F*) ⊂ Aut_K(F) and φ with φ(F*) ⊂ Ant_t(F) is defined — such a product we will call a (K · t)-coupling. If κ is a (K · t)-coupling, Char K ≠ 2 and the image of κ is not Klein’s 4-group, then F ^κ ≅ F ^κ′, where κ′ has an especial form; in fact, there are four different possibilities for κ′. We explicitly determine all the possibilities for κ′ and show how such couplings κ′ can be constructed.