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Extensions of locally finite André systems. (Erweiterungen lokal endlicher André-Systeme.) (German) Zbl 0783.51003
If \((F,+,\cdot)\) is a field then \(\kappa:F\backslash\{0\}=F^*\to\operatorname{Aut}(F,+,\cdot)\); \(x\to\kappa_ x\) is called a coupling if \(\forall a,b\in F\backslash\{0\}:\kappa_ a\circ\kappa_ b=\kappa_{a\cdot\kappa_ a(b)}\). Then \((F,+,\circ)\) with \(a\circ b:=a\cdot\kappa_ a(b)\) is a nearfield, called Dickson nearfield [cf. e.g. H. Wähling, Theorie der Fastkörper (1987; Zbl 0669.12014)]. This can be generalized: Let \(K\leq F\) and \(\varphi:F^*\to GL(F,K)\); \(x\to\varphi_ x\) such that \(\varphi_ 1=id\) and \(\forall a,b\in F^*\), \(\varphi_ a(1)=1\) and \(\exists_ 1x\in F^*:x\varphi_ x(a)=b\). Here \(F^ \varphi:=(F,+,\circ)\) with \(a\circ b:=a\cdot\varphi_ a(b)\) is only a quasifield.
The author continues his studies on André-systems [cf. Arch. Math. 58, No. 5, 514-520 (1992; Zbl 0725.12006) and J. Geom. 41, No. 1/2, 79-93 (1991; Zbl 0734.51003)] and considers here the particular case that \((Q,K)\) is an extension of locally finite fields (an algebraic structure is called locally finite, if any finite subset is contained in a finite substructure) and that \(\varphi=\psi\circ N\) has a factorization where \(N:Q^*\to(K^*,\cdot)\) is a homomorphism and \(\psi:N(Q^*)\to\operatorname{Aut}(Q,K)\) a map with \(\psi(1)=id\). Then he calls \(\varphi\) and André-derivation and \(Q^ \varphi\) an André-system. He studies the possibility to extend a finite André-system to a countable locally finite André-system. Using Steinitz-numbers he gives a complete survey on the extensibility of finite André-derivations \(\varphi_ 0:Q^*_ 0\to\operatorname{Aut}(Q_ 0,K_ 0)\) to André-derivations \(\varphi:Q^*\to\operatorname{Aut}(Q,K)\) where \(Q\) is firstly a finite extension field of \(Q_ 0\) (Theorem 1.3) and secondly a locally finite extension field (Theorem 2.1), and on the cardinality of nonisomorphic quasifields \(Q^ \varphi\) (Theorem 2.3).
MSC:
51A25 Algebraization in linear incidence geometry
51A35 Non-Desarguesian affine and projective planes
51A40 Translation planes and spreads in linear incidence geometry
16Y30 Near-rings
12K05 Near-fields
12K99 Generalizations of fields
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