By Serge Lang (auth.)
This ebook is meant as a simple textual content for a one-year direction in Algebra on the graduate point, or as an invaluable reference for mathematicians and pros who use higher-level algebra. It effectively addresses the fundamental techniques of algebra. For the revised 3rd version, the writer has additional workouts and made a number of corrections to the text.
Comments on Serge Lang's Algebra:
Lang's Algebra replaced the best way graduate algebra is taught, holding classical subject matters yet introducing language and methods of pondering from class concept and homological algebra. It has affected all next graduate-level algebra books.
April 1999 Notices of the AMS, saying that the writer was provided the Leroy P. Steele Prize for Mathematical Exposition for his many arithmetic books.
The writer has a magnificent knack for featuring the $64000 and engaging principles of algebra in exactly the "right" means, and he by no means will get slowed down within the dry formalism which pervades a few components of algebra.
MathSciNet's overview of the 1st edition
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This 5th variation of Lang's e-book covers all of the issues typically taught within the first-year calculus series. Divided into 5 components, each one element of a primary direction IN CALCULUS comprises examples and functions in terms of the subject lined. additionally, the rear of the publication comprises precise recommendations to a great number of the workouts, permitting them to be used as worked-out examples -- one of many major advancements over prior variants.
Extra resources for Algebra
For each x E G we define the translation Tx : G � G by Tx( Y) = xy . Then the map (x, y) �----+ xy = �(y) defines an operation of G on itself. Warning : Tx is not a group-homomorphism ! Only a permutation of G. OPERATIONS OF A G ROU P ON A SET I, §5 27 Similarly, G operates by translation on the set of subsets, for if A is a subset of G, then x A = Yx(A) is also a subset. If H is a subgroup of G, then Tx(H) = xH is in general not a subgroup but a coset of H, and hence we see that G operates by translation on the set of co sets of H.
E . the subgroup of x E A such that mx = 0 . We prove : If m = rs with r, s positive relative prime integers, then A m = A r + A s . Indeed , there exist integers u, v such that ur + vs = 1 . Then x = urx + vsx, and urx E A s while vsx E An and our assertion is proved . Repeating this process inductively , we conclude: If m = Il p e( p ) then A m = L Ap e(p). pl m plm Hence the map ffi A(p) � A is surjective, and the theorem is proved . Let A = Q/Z . Then Q/Z is a torsion abelian group , isomorphic to the direct sum of its subgroups (Q/Z)(p) .
Throughout this section we write our abelian groups additively. Let A be an abelian group. An element a E A is said to be a torsion element if it has finite period. The subset of all torsion elements of A is a subgroup of A called the torsion subgroup of A. ) The torsion subgroup of A is denoted by A t on or simply A , . An abelian group is called a torsion group if A = A tor , that is all elements of A are of finite order. A finitely generated torsion abelian group is obviously finite. We shall begin by studying torsion abelian groups.
Algebra by Serge Lang (auth.)