Title
Quadratische Formen und orthogonale Gruppen (Grundlehren der mathematischen Wissenschaften, 63) (German Edition),Used
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wirkung des Raumes mit seiner Bewegungsgruppe. Dementsprechend ist der Weg, auf dem del' Leser hier gefiihrt wird, zweigleisig; es wechseln Dbedegungen, welche den Raum bzw. die seine Metrik definierende quadratische Form betreffen, mit Betrachtungen liber seine Bewegungs gruppe, die orthogonale Gruppe im weitesten Sinne. Der Titel bringt diese doppelte Aufgabe zum Ausdruck. Beachtet man, daB die Theorie der hyperkomplexen Systeme in ihrer historischen Entwicklung und ihrem heutigen Bestand weitgehend mit der Darstellung von Gruppen durch Abbildungen eines affinen Raumes auf sich ubereinstimmt, so ergibt sich damit die Stellung im heutigen GefUge der Mathematik, welche die Arithmetik der quadratischen Formen beanspruchen muB. Sie ist im gleichen Sinne neben del' hyperkomplexen Algebra und Arith metik einzuordnen, wie die orthogonale Gruppe neben der affinen steht. Ich hoffe, daB die Herausarbeitung der gruppentheoretischen Motive in del' Theorie der quadratischen Formen den Erfolg hat, daB die beiden aus den Disquisitiones Arithmeticae erwachsenen Zweige der Arithmetik einander naher gebracht werden, und daB so die Einheit un serer Wissen schaft gefordert wird. Wenngleich das Buch vieles in diesel' Form Neue bringt, bin ieh mil' bewuBt, daB mil' die Anregungen hierzu von vielen Seiten zugeflossen sind, wovon die im Text vorkommenden Namen, die vielfach un serer Generation angehoren, Zeugnis ablegen. Nicht immer ist es aber moglich, den Urheber eines Gedankens exakt festzulegen; Wissenschaft ist Gemeinschaftsar bei t.
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