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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.

12K99 Generalizations of fields
Full Text: DOI
[1] J. André, Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Math. Z.60, 156-186 (1954). · Zbl 0056.38503 · doi:10.1007/BF01187370
[2] R. H.Bruck, A survey of binary systems, 2nd Printing. Berlin-Heidelberg-New York 1966. · Zbl 0141.01401
[3] A. Caggegi andA. Herzer, The generalized André systemsA(F, ?, (g i), (f i),?). Abh. Math. Sem. Univ. Hamburg58, 219-236 (1988). · Zbl 0695.51001 · doi:10.1007/BF02941379
[4] D. A. Foulser, A Generalization of Andre’s Systems. Math. Z.100, 380-395 (1967). · Zbl 0152.18903 · doi:10.1007/BF01110421
[5] H. Hähl, Achtdimensionale lokalkompakte Translationsebenen mit großer Streckungsgruppe. Arch. Math.34, 231-242 (1980). · Zbl 0443.51010 · doi:10.1007/BF01224957
[6] A. Herzer, Zu einem Satz von H. Lüneburg über verallgemeinerte André-Ebenen. Arch. Math.52, 99-104 (1989). · Zbl 0633.51003 · doi:10.1007/BF01197979
[7] D. R.Hughes and F. C.Piper, Projective Planes. Berlin-Heidelberg-New York 1973.
[8] H.Lüneburg, Translation Planes. Berlin-Heidelberg-New York 1980.
[9] S. J. Tillman, The Multiplicative Group of Absolutely Algebraic Fields in Characteristicp. Proc. Amer. Math. Soc.23, 601-604 (1969). · Zbl 0188.11102
[10] J. Timm, Zur Konstruktion von Fastringen I. Abh. Math. Sem. Univ. Hamburg35, 57-74 (1970). · Zbl 0217.06402 · doi:10.1007/BF02992475
[11] H. Wähling, Projektive Inzidenzgruppoide und Fastalgebren. J. Geom.9, 109-126 (1977). · Zbl 0351.50009 · doi:10.1007/BF01918063
[12] H. Wähling, Normale Teilquasikörper eines Fastrings. Der Satz von Cartan-Brauer-Hua. Math. Z.158, 55-60 (1978). · Zbl 0381.16020 · doi:10.1007/BF01214565
[13] H.Wähling, Theorie der Fastkörper. Essen 1987. · Zbl 0669.12014
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