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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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##### References:
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