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4, Satz 31) hat gezeigt: Die addi ti ve Gruppe j edes planaren Quasikorpers ist kom- ( 4 . 5) mutativ. 0 (K,+,·) sei ein planarer Quasikorper. Z(K,+,·) :={aEK*1 ax=xa flir aIle xEK} +ya und Wenn heiBt Zentrum und Kern(K,+,·):={aEKI(x+y)a=xa+ (xy)a=x(ya) fliralle x,YEK} heiBtKernvon (K,+,·). Z(K,+,·) = K* gilt, heiBt (K,+,·) korrunutativ. J. 6 ) Der Kern j edes planaren Quasikorpers ist ein Korper. Jeder planare Quasikorper ist ein Rechtsvektorraum tiber seinem Kern. 6) (vgl. 4): (K,+,·) sei ein planarer Quasikorper.

Beispielsweise [55], S. 249, [65], S. 117): a+b :=T(a,l,b) fUr a,b E K, a . b:= T(a,b,O) fUr a,b E K. Die VerknUpfung + heiSt Addition und die VerknUpfung . Multiplikation. Statt a·b wird kurz ab und statt a·a kurz a geschrieben. 2 Ein Ternarkorper (K,T) heiSt linear, wenn T(a,b,c) = a . b + C fUr aIle a,b,c E K gilt. 1) folgt unmittelbar (vgl. beispielsweise [ 55], S. 249, [65], Thm. 5. 2) FUr einen Ternarkorper (K,T) mit Addition + und Multipli- kat ion . gel ten: (1) (K,+) ist eine Loop mit neutralem Element 0.

11 ~2 mit } ~ki-O. 1 1 ,J 12 daher R(p)e~(K,+,·)\lI(K,+,·) Fall 2: Es gilt b~ 12 und p. ~lt ~2 < -~. ~i ~ ~ und p . =0 fUr jel\{illk} gilt also 1eR(p), und ~(p,b~ )=~. +p. =b. gilt 11 11 111 peP mit P, . =1, ...... 1,1 1 ~(p,b. )=p . 17)(2). ~ . FUr jedes 11 ~(p,b. ) ~i ~i b. ~ . = p. peP mit p . =1, p . <-1 1 1 ,1 1 . (p,b. )=p . (p,l+b. )=p . +p . (p,b. +b~)= 12 = p. 1 1 1 ,1 1 1 ,1 2 12 S· +p . 1 11 1 ,1 2 < 0, also 1 1 ,1 1 1 1 ,1 2 12 12 b. ,1+b. ,b. (l+b. 17)(1). Die Anzahl der Abbildungen PEP, die die in Fall 1 bzw.

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