# Download e-book for kindle: Algebra II: Chapters 4-7 (Pt.2) by Nicolas Bourbaki

By Nicolas Bourbaki

ISBN-10: 3540193758

ISBN-13: 9783540193753

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Additional info for Algebra II: Chapters 4-7 (Pt.2)

Sample text

NO. 45 Hence the algebra TS(M) is commutative. It is to be noted that the canonical injection of TS (M) in T(M) is not in general an algebra homomorphism. Worse still, TS(M) is not in general stable under the multiplication of T (M ). 4. Divided powers Let x E M and k E N. It is clear that x, @ x, 8 ... @ xk, where is an element of T S ~ ( M ) . DEFINITION 2. - If x E M, the element x Q x Q ... Q x of T S ~ ( M )is denoted by Y~(x). PROPOSITION 3. - (i) If x E M, the pth power of x, calculated in TS ( M ), is equal to P !

3 of IV, p. 33 we have whence I (e(X)) = X. Let K be a Q-algebra, then the elements of K[[I]] without constant term form a commutative group d under addition. The elements of K [[I]] with constant term 1 form a commutative group A under multiplication (IV, p. 30). For each f E 8 , we can define the elements e o f and I o f of d , and by Prop. 14 above, the mappings f H I o f and f H e o f are mutually inverse permutations of 8 ; clearly they are continuous. Since exp X = e ( X ) + 1, we see that the exponential By mapping f H exp f = e 0 f + 1 is a continuous bijection of d onto 4.

Pn. , p ) into (1, ... , c a r d cp-'(n) = p , . Then Y,,(xI) ~ ~ ~ (YPn(xn) ~ 2 = 1 C Xq(1) 0' ~ ( 2 ) 0 ... @ X q @ ) . ,x, E 3 0. , n ) The assertion (i) follows at once from Prop. 2 (ii). r,(xH). 46 95 POLYNOMIALS AND RATIONAL FRACTIONS Let us prove (ii) ; by an induction on n we see that it is enough to consider the case n = 2. Then we have yp(xl+x2)= (x, + x 2 ) Q ( x l + x 2 ) Q ... Q ( x I + x 2 ) @factors) To prove (iii), let Gpl,,,,,pn be the set of permutations of (1, p, restrictions to the intervals + .