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The kernel of an automorphic derivation and an application to normal subfields of generalized André-systems. (Der Kern einer automorphen Ableitung und eine Anwendung auf normale Teilkörper verallgemeinerter André-Systeme.) (German) Zbl 0725.12006
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 $$\operatorname{Aut}(Q)$$, then $$\phi$$ is called automorphic and $$Q^{\phi}$$ is usually named a generalized André-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 [D. A. Foulser, Math. Z. 100, 380–395 (1967; Zbl 0152.18903); 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 André-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. H. Wähling, Theorie der Fastkörper. Essen: Thales Verlag (1987; Zbl 0669.12014), (III.5.5)]. In the last section some examples are given.

##### MSC:
 12K99 Generalizations of fields
Full Text:
##### References:
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