an:04196099 Zbl 0725.12006 Kiechle, Hubert The kernel of an automorphic derivation and an application to normal subfields of generalized Andr??-systems DE Arch. Math. 58, No. 5, 514-520 (1992). 00011193 1992
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12K99 derivation; generalized Andr??-system; kernel; nuclei; fixed field; center of the derived quasifield; normal subfields of nearfields Let $$Q$$ be a quasifield and $$K$$ a subfield (not necessary commutative) of the kernel of $$Q$$. A map $$\phi: Q^*\to \mathrm{GL}(Q,K)$$; $$a\to \phi_ a$$ $$(Q^*:=Q\setminus \{0\})$$ is called a derivation, if the derived quasifield $$Q^{\phi}:=(Q,+,\circ)$$ is a quasifield as well. Here $$a\circ b:=a\phi_ a(b)$$, $$a\neq 0$$ and $$0\circ b:=0$$. The subgroup of $$\mathrm{GL}(Q,K)$$ generated by $$\phi (Q^*)$$ is denoted by $$\Delta_{\phi}$$. If $$Q$$ is a (skew)field and $$\Delta_{\phi}$$ is contained in $$\Aut(Q)$$, then $$\phi$$ is called automorphic and $$Q^{\phi}$$ is usually named a generalized Andr??-system. The kernel of $$\phi$$ is the set $$\mathrm{Ker}\,\phi:=\{a\in Q^*;\;\phi_{ax}=\phi_ x\}$$, and is a subgroup of $$Q^*$$. This idea implicitly has been used before [\textit{D. A. Foulser}, Math. Z. 100, 380--395 (1967; Zbl 0152.18903); \textit{A. Herzer}, Arch. Math. 52, No. 1, 99--104 (1989; Zbl 0633.51003)]. It proved very useful in the study of the structure of generalized Andr??-systems. Some relations between $$\mathrm{Ker}\,\phi$$ and the nuclei, the fixed field of $$\Delta_{\phi}$$ and the center of the derived quasifield are given. These results are used to generalize theorems on normal subfields of nearfields [cf. \textit{H. W??hling}, Theorie der Fastk??rper. Essen: Thales Verlag (1987; Zbl 0669.12014), (III.5.5)]. In the last section some examples are given. Hubert Kiechle (M??nchen) Zbl 0633.51003; Zbl 0669.12014; Zbl 0152.18903