By Wolfgang Walter (auth.)

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An=b 1 + ... +b n. Dabei bilden die bi eine Umordnung der ai , wenn eine Bijektion

0 Bemerkung. Bis ins 19. Jahrhundert verstand man unter Algebra im wesentlichen das Umgehen mit Polynomen und insbesondere die Bestimmung von Polynomwurzeln. Die Entdeckung oder, wenn man so will, die Schaffung der komplexen Zahlen ist aufs engste mit dieser Aufgabe verknüpft. Der sogenannte Fundamentalsatz der Algebra, auf den wir in §8 zurückkommen werden, sagt aus, daß ein Polynom vom Grad n genau n komplexe NullstelIen hat. Seine Geschichte ist im Grundwissen-Band Zahlen dargestellt. Durch mehrfache Anwendung von (c) zeigt man ohne Schwierigkeit, daß ein Polynom vom Grad n eine Darstellung von der Form P(x)=(x - ~ l)(X - ~2)'" (x - ~k) Q(x) besitzt, wobei ksn und Q(x) ein Polynom vom Grad n-k ist, welches keine reellen Nullstellen hat.

Transzendente Zahlen. Jede nicht-algebraische reelle Zahl wird transzendent genannt. 10 zeige man, daß jedes Intervall abzählbar viele algebraische und überabzählbar viele transzendente Zahlen enthält. 3. Fibonaccische Zahlen. +2-l. i=O Bei (a) geht die Summe nur bis [nI2], bei (b) benutze man die Binetsche Darstellung. 4. Lucassche und verwandte Zahlen. 17 unter der Anfangsbedingung ao=a, a 1=b, wobei a, b reelle Zahlen sind. Im Fall (a,b)=(2,1) ergeben sich die sog. Lucasschen Zahlen Ln: Ln+1=Ln+Ln_1 ftir n=I,2, ...

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