Algebra: From Undergrad to Grad | Serge Lang

[micromass]

• Author: Serge Lang
• Title: Algebra
• Amazon Link: https://www.amazon.com/dp/038795385X/?tag=pfamazon01-20
• Prerequisities: Good knowledge of undergrad Algebra
• Level: Grad

Table of Contents:

[LIST]
[*] The Basic Objects of Algebra
[LIST]
[*] Groups 
[LIST]
[*] Monoids
[*] Groups
[*] Normal subgroups
[*] Cyclic groups
[*] Operations of a group on a set 
[*] Sylow subgroups 
[*] Direct sums and free abelian groups
[*] Finitely generated abelian groups 
[*] The dual group
[*] Inverse limit and completion 
[*] Categories and functors
[*] Free groups
[/LIST]
[*] Rings 
[LIST]
[*] Rings and homomorphisms
[*] Commutative rings
[*] Polynomials and group rings
[*] Localization
[*] Principal and factorial rings
[/LIST]
[*] Modules
[LIST]
[*] Basic definitions
[*] The group of homomorphisms
[*] Direct products and sums of modules
[*] Free modules
[*] Vector spaces
[*] The dual space and dual module
[*] Modules over principal rings
[*] Euler-Poincare maps
[*] The snake lemma
[*] Direct and inverse limits
[/LIST]
[*] Polynomials
[LIST]
[*] Basic properties for polynomials in one variable
[*] Polynomials over a factorial ring
[*] Criteria for irreducibility
[*] Hilbert's theorem
[*] Partial fractions
[*] Symmetric polynomials
[*] Mason-Stothers theorem and the abc conjecture 
[*] The resultant
[*] Power series
[/LIST]
[/LIST]
[*] Algebraic Equations
[LIST]
[*] Algebraic Extensions 
[LIST]
[*] Finite and algebraic extensions
[*] Algebraic closure 
[*] Splitting fields and normal extensions
[*] Separable extensions
[*] Finite fields
[*] Inseparable extensions
[/LIST]
[*] Galois Theory 
[LIST]
[*] Galois extensions
[*] Examples and applications
[*] Roots of unity
[*] Linear independence of characters
[*] The norm and trace
[*] Cyclic extensions
[*] Solvable and radical extensions
[*] Abelian Kummer theory
[*] The equation X^n - a = 0
[*] Galois cohomology
[*] Non-abelian Kummer extensions
[*] Algebraic independence of homomorphisms 
[*] The normal basis theorem
[*] Infinite Galois extensions
[*] The modular connection
[/LIST]
[*] Extensions of Rings
[LIST]
[*] Integral ring extensions
[*] Integral Galois extensions
[*] Extension of homomorphisms
[/LIST]
[*] Transcendental Extensions
[LIST]
[*] Transcendence bases
[*] Noether normalization theorem
[*] Linearly disjoint extensions
[*] Separable and regular extensions
[*] Derivations
[/LIST]
[*] Algebraic Spaces 
[LIST]
[*] Hilbert's Nullstellensatz
[*] Algebraic sets, spaces and varieties 
[*] Projections and elimination
[*] Resultant systems
[*] Spec of a ring
[/LIST]
[*] Noetherian Rings and Modules
[LIST]
[*] Basic criteria
[*] Associated primes
[*] Primary decomposition
[*] Nakayama's lemma
[*] Filtered and graded modules
[*] The Hilbert polynomial
[*] Indecomposable modules
[/LIST]
[*] Real Fields 
[LIST]
[*] Ordered fields
[*] Real fields
[*] Real zeros and homomorphisms 
[/LIST]
[*] Absolute Values
[LIST]
[*] Definitions, dependence, and independence
[*] Completions
[*] Finite extensions
[*] Valuations
[*] Completions and valuations
[*] Discrete valuations
[*] Zeros of polynomials in complete fields
[/LIST]
[/LIST]
[*] Linear Algebra and Representations
[LIST]
[*] Matrices and Linear Maps 
[LIST]
[*] Matrices
[*] The rank of a matrix
[*] Matrices and linear maps
[*] Determinants
[*] Duality
[*] Matrices and bilinear forms
[*] Sesquilinear duality
[*] The simplicity of SL_2(F)/\pm 1
[*] The group SL_n(F),n\geq 3
[/LIST]
[*] Representation of One Endomorphism
[LIST]
[*] Representations
[*] Decomposition over one endomorphism
[*] The characteristic polynomial
[/LIST]
[*] Structure of Bilinear Forms
[LIST]
[*] Preliminaries, orthogonal sums
[*] Quadratic maps
[*] Symmetric forms, orthogonal bases
[*] Symmetric forms over ordered fields
[*] Hermitian forms
[*] The spectral theorem (hermitian case)
[*] The spectral theorem (symmetric case)
[*] Alternating forms
[*] The Pfaffian
[*] Witt's theorem
[*] The Witt group
[/LIST]
[*] The Tensor Product
[LIST]
[*] Tensor product
[*] Basic properties
[*] Flat modules
[*] Extension of the base
[*] Some functorial isomorphisms
[*] Tensor product of algebras
[*] The tensor algebra of a module
[*] Symmetric products
[/LIST]
[*] Semisimpliclty
[LIST]
[*] Matrices and linear maps over non-commutative rings
[*] Conditions defining semisimplicity
[*] The density theorem
[*] Semisimple rings
[*] Simple rings
[*] The Jacobson radical, base change, and tensor products
[*] Balanced modules
[/LIST]
[*] Representations of Finite Groups
[LIST]
[*] Representations and semisimplicity
[*] Characters
[*] 1-dimensional representations
[*] The space of class functions
[*] Orthogonality relations
[*] Induced characters
[*] Induced representations
[*] Positive decomposition of the regular character
[*] Supersolvable groups
[*] Brauer's theorem
[*] Field of definition of a representation
[*] Example: GL_2 over a finite field
[/LIST]
[*] The Alternating Product
[LIST]
[*] Definition and basic properties
[*] Fitting ideals
[*] Universal derivations and the de Rham complex 
[*] The Clifford algebra
[/LIST]
[/LIST]
[*] Homological Algebra
[LIST]
[*] General Homology Theory
[LIST]
[*] Complexes
[*] Homology sequence
[*] Euler characteristic and the Grothendieck group
[*] Injective modules
[*] Homotopies of morphisms of complexes
[*] Derived functors
[*] Delta-functors
[*] Bifunctors
[*] Spectral sequences
[/LIST]
[*] Finite Free Resolutions 
[LIST]
[*] Special complexes
[*] Finite free resolutions
[*] Unimodular polynomial vectors
[*] The Koszul complex
[/LIST]
[/LIST]
[*] Appendix: The Transcendence of e and \pi
[*] Appendix: Some Set Theory
[*] Bibliography
[*] Index
[/LIST]

 

[mathwonk]

I'm not saying this is a good book to learn from, but my algebraic friends do say at some point one should aspire to knowing what is in it. So use it a a goal, and try to find something else that let's you reach that goal. maybe start with dummit and foote or artin, and move up to lang.

caveat" there are almost no problems, so i recommend combining this book with hungerford, or dummitt and foote for problems.

i.e. YOU CANNOT LEARN FRON LANG ALONE! no problems means no understanding.

 

[A. Bahat]

I really like this book.

There are relatively few examples and problems in Lang. However, when I think of this book, I think of the examples and problems as perhaps the best part! Here's why: although they are fewer in number, they are excellent. In the examples, you can find compact Riemann surfaces, algebras of continuous functions, covering spaces, fundamental groups, spectral theory, affine schemes, Picard groups, and on and on! Lang's breadth is apparent in this book, and nothing pitched at a lower level will give you such rich connections with other fields of mathematics. Needless to say, this necessitates some more maturity on the reader's part, but it will be amply rewarded.

The problems, too, are great: Bernoulli polynomials, Dedekind rings, linear fractional transformations, "take any book on homological algebra and prove all the theorems without looking at the proofs"... They don't seem like "exercises" which are just meant to practice some little skill needed for a course--you get the feeling of doing real mathematics. Many are open-ended, many contain references to the literature; in short, this is a perfect book for the aspiring mathematician transitioning from student-hood to research.

Warning: typos are plentiful, as are careless mistakes and sloppy explanations, but I think that the book as a whole is still remarkably valuable.

 

[jbunniii]

Anyone trying to read Lang would do well to check out this very nice Companion to Lang's Algebra:

http://math.berkeley.edu/~gbergman/.C.to.L/

 

[mathwonk]

my formative copy of lang was the first edition. there are many more problems in the later ones, but still probably not enough examples. on the other hand there are many things in lang that other books slight. when i was in grad school, lang was the only book to fill in the theory our courses took for granted. i would say for a future mathematician lang is necessary but not sufficient.

as a supplement hungerford has a lot of good examples, and i personally would suggest using it in preference to the notes by bergman, which to me, with all respect, seem tedious and tend to make the extreme formality and abstraction of lang even worse. they obviously will have appeal to some, and i say this only for those like me who have a different perspective.