Algebra II: Textbook for Students of Mathematics by Alexey L. Gorodentsev PDF

By Alexey L. Gorodentsev

ISBN-10: 3319508520

ISBN-13: 9783319508528

ISBN-10: 3319508539

ISBN-13: 9783319508535

This publication is the second one quantity of a radical “Russian-style” two-year undergraduate path in summary algebra, and introduces readers to the elemental algebraic buildings – fields, earrings, modules, algebras, teams, and different types – and explains the most ideas of and techniques for operating with them.

The direction covers giant components of complex combinatorics, geometry, linear and multilinear algebra, illustration concept, class concept, commutative algebra, Galois concept, and algebraic geometry – subject matters which are usually missed in average undergraduate courses.

This textbook is predicated on classes the writer has performed on the self sustaining college of Moscow and on the school of arithmetic within the greater tuition of Economics. the most content material is complemented via a wealth of workouts for sophistication dialogue, a few of which come with reviews and tricks, in addition to difficulties for self sustaining study.

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Extra resources for Algebra II: Textbook for Students of Mathematics

Example text

N 1/ . n 1/-linear form on V obtained from the form ' by setting the first argument equal to v. v; u1 ; u2 ; : : : ; un 1 / for all u1 ; u2 ; : : : ; un 1 2 V. Indeed, since both sides of the equality are linear in ', it is enough to verify it only for the n-linear forms ' coming from the decomposable tensors 1 because the latter span V ˝n ˝ 2 ˝ ˝ n 2V ˝n ; . U \ W/˝n in V ˝n . 3 that for every tensor t 2 V ˝n , the intersection of all vector subspaces U V such that t 2 U ˝n is the minimal subspace of V with respect to inclusions whose nth tensor power contains t.

2 Combinatorial Formula for Complete Polarization Since the value of a symmetric n-linear form does not depend on the order of arguments, let us write e f v1m1 ; v2m2 ; : : : ; vnmk for the value of e f at m1 vectors v1 , m2 vectors v2 ; : : : ; mk vectors vk with m D n. 28) where the summation is over all integers m1 ; m2 ; : : : ; mk such that m1 C m2 C C mk D n and 0 6 m 6 n for all . 6 Let V be a vector space, not necessarily finite-dimensional, over a field  of characteristic zero. 29) . v1 ; v2 ; : : : ; vn / D I¨f1;:::;ng i…I where the left summation is over all subsets I ¨ f1; 2; : : : ; ng including I D ¿, for which j¿j D 0.

V; W/; F 7! F ∘ ': A symmetric multilinear map ' is called universal if %' is an isomorphism for all W. The universal symmetric n-linear map is also called the n-ary commutative multiplication of vectors. 7 Verify that the target spaces of any two universal symmetric nlinear maps are isomorphic by means of the unique linear map commuting with the commutative multiplication. , Proof By the universal property of tensor multiplication W V V V ! V ˝n , every n-linear map ' W V V V ! W is uniquely factorized as ' D e F∘ for some linear map e F W V ˝n !

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Algebra II: Textbook for Students of Mathematics by Alexey L. Gorodentsev

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