Algebra: A Text-Book of Determinants, Matrices, and by W. L. Ferrar

By W. L. Ferrar

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Let H =A (x~x 2) over K 2 w i t h d~ I = l, d~ = 2, by h(x l) = x I @ 1 + 1 ~ Xl, h(x 2) = x 2 @ 1 + 1 ~ x 2 + x I ~ x 1. T h e n H is i s o m o r p h i c to H 1 | H 2 where H 1 = A (x l) w i t h hl(X l) = x I ~ 1 + 1 9 Xl, H 2 = A (x 2) w i t h h 2 ( x 2) = x 2 ~ 1 + 1 9 x 2. However Thus there an a l g e b r a Let h(x) is c l e a r l y can have of a system particular We essentially x r H be a h o m o g e n e o u s = x | 1 + 1 | x. One existence no i s o m o r p h i s m case where sees distinct element readily of p r i m i t i v e homomorphlsms w i t h d~ that h onto h I | h 2.

F r o m the above be c a n c e l l e d We are now But t h e n r d O X k < n. n and k ~ O, it follows P = O. ,x l) is proved. ,Xl) z is a m o n o m i a l The p r o o f that p ~ 0 then by an a r g u m e n t in the p r o o f normal (x i) is a m i n i m a l system. Thus the normal monomials independent. characteristic recalled Every fact = x k + R ( X k , . . , x l) = O. monomial (not n e c e s s a r i l y is by i n d u c t i o n similar to of t h e o r e m r > 1 and is a power (B) of p. 1. in P can be w r i t t e n in the f o r m z r where ~e Kp and normal).

II, Paris (1949-5o). [51 H. Cartan, idem III, (195o-51). [61 S. Eilenberg, Idem III, (195o-51). [71 T. Kudo, Homological structure of fibre bundles, Jour. Inst. of Polyt. Osaka City Univ. 2 (1952), lol-14o. [81 J. Leray, Espaces o~ op~re un groupe de Lie compact connexe, Comptes rendus de l'Acad~mle des Sciences, Paris, t. 228 (1949), pp. 1545-47. , pp. 1784-86. - [9] J. Math. Pures Appl. IXs. 29 (195o), 169-213. D. Liao, On the theory of obstructions of fibre bundles, Ann. of Math. (1954), 146-191.

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