×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.