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By Jean Dieudonné

Written by means of a world-renowned mathematician, this good educated and vintage textual content lines the background of algebraic topology, starting with its production within the early 1900s. It is going directly to describe intimately the real theories that have been stumbled on prior to 1960.

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A unabhangig voneinander ein Reprasentantensystem in K mod. rr durchlaufen sollen. Hier liegt der i-te Faktor H - a ^ / i V in dem kommutativen Teilkorper K^uJJ") von D. , so folgt unser Satz ohne Schwierigkeit. A m Schluss bemerken w i r noeh, dass die Satze 3, 4 und 4' sich auch auf den Fall einer einfachen Algebra iibertragen lassen. Man erreicht dies leicht mit Hilfe der Theorie der Elementarteilern und der bekannten Struktur der additiven Gruppe der ganzen Elemente aus D \ h F H 8) V g l .

MATSUSHIMA. Kibtf) zu setzen und zu bemerken, dass die ganzen rationalen p-adischen Potenzen der Elementen aus einem Korper K wieder in K liegen. 5. Auch der obige Satz 4 bleibt giiltig fiir Divisionsalgebren der in der Bemerkung in § 3 betrachteten A r t . Dies folgt unmittelbar aus dem folgenden allgemeinen Satz U' • K sei ein beliebiger diskret bewerteter perfekter Korper, und D eine Divisionsalgebra iiber K. ,K„ in D, so dass jede Einseinheit e in D mit Einseinheitene e< aits Ki sich midtiplikativ in der Form { 2 e = e ...

S From this we see easily that F„ has the following form : Since B is evidently a replica of A, , we have from (5), (7) and (8) the following equations : n s ,M0 =-m (*=T, 2, —MH0 t ! 5 , ^ ( 0 = i ' « , j - f - 7 ( j + ^ ( j * t + ••• + 7r»j&(/», { BJF. g) = vF„, F , a . Hence we have B,,F. = vF« = B 'F* and B»F. = ls B,/F.. ,. From these lemmas we may prove the following Theorem. /) are arbitrry elements of the field K. Proof. The sufficiency may be seen from the Lemma 2 and 3 . Let, conversely, B be a replica of A.

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