## 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.

Original language | English |
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Pages (from-to) | 289-311 |

Number of pages | 23 |

Journal | Results in Mathematics |

Volume | 44 |

Issue number | 3-4 |

DOIs | |

State | Published - 1 Nov 2003 |

## Keywords

- 12F20
- coupling
- simple transcendental extension