By Nicolas Bourbaki (auth.)

This is a softcover reprint of the English translation of 1990 of the revised and improved model of Bourbaki's, Algèbre, Chapters four to 7 (1981).

This completes Algebra, 1 to three, via setting up the theories of commutative fields and modules over a valuable excellent area. bankruptcy four bargains with polynomials, rational fractions and tool sequence. a piece on symmetric tensors and polynomial mappings among modules, and a last one on symmetric features, were further. bankruptcy five used to be fullyyt rewritten. After the fundamental conception of extensions (prime fields, algebraic, algebraically closed, radical extension), separable algebraic extensions are investigated, giving solution to a bit on Galois conception. Galois conception is in flip utilized to finite fields and abelian extensions. The bankruptcy then proceeds to the research of common non-algebraic extensions which can't frequently be present in textbooks: p-bases, transcendental extensions, separability criterions, general extensions. bankruptcy 6 treats ordered teams and fields and in response to it really is bankruptcy 7: modules over a p.i.d. stories of torsion modules, unfastened modules, finite style modules, with purposes to abelian teams and endomorphisms of vector areas. Sections on semi-simple endomorphisms and Jordan decomposition were added.

Chapter IV: Polynomials and Rational Fractions

Chapter V: Commutative Fields

Chapter VI: Ordered teams and Fields

Chapter VII: Modules Over primary perfect Domains

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Extra info for Algebra II: Chapters 4–7

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Xn E (q + r)! " "Yq+r(x), q. r. M, and for He {l, ... , n} put L Xi' XH = i He {I .... } The assertion (i) follows at once from Prop. 2 (ii). 46 §5 POLYNOMIALS AND RATIONAL FRACfIONS 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(XI + X2) = (Xl + X2) ® L (Xl L + X 2 ) ® ... ® a(xI (Xl + X2) (p factors) ® Xl ® ... ® Xl ® X2 ® X2 ® ... ® X2) PI+PZ~P aES pl • PZ L L a(-YPI(XI) ® "YPz (X2)) PI + PZ ~ P a E SPI' P2 L PI +PZ "YPI (Xl) "YPZ (X2 ) • ~P To prove (iii), let 6 pl , ...

We recall (III, p. 501) that 6 n operates on the left of the A-module Tn(M), in such a way that a(x J ® X2 ® ... ® xn) = X,,-l(l) ® X,,-1(2) ® ... ® X,,-l(n) for any Xl' ... , Xn EM and a E 6 n. The elements Z E Tn(M) such that a • Z = Z for all a E 6 n are called symmetric tensors of order n ; they form a sub-A-module of Tn(M) denoted by Tsn(M); we have TSo(M) = A, TSJ(M) = M. 43 00 TS(M) = EB Tsn(M); this is a graded sub-A-module of T(M). For every n=O Z E Tn (M) the element. LIT. z belongs to TS n (M); we denote it by s .

U . L xl ® Xz ® ... ® x~ be an element of TSn(M), where the xj belong . p.. Let v = i = . 1 to M; then I/IM(v) is equal to P L xfx~ ... x~ calculated in SCM), whence i = 1 L s(xf ® x~ ® ... v . 54 COROLLARY 1. - If A is a Q-algebra, then the canonical homomorphism of 5 (M) into TS (M) is an algebra isomorphism. If moreover M is free, then it is an isomorphism of graded bigebras. COROLLARY 2. - If A is a Q-algebra then the module Tsn(M) is generated by the n-th powers of elements ofM in TS(M). This follows from Cor.

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