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Abstract Algebra

Groups, rings, fields: symmetry made into structure.

经验教训

Introductory Groups The axioms, examples (integers mod n, symmetries, permutations), order of an element. 3^100 mod 7 Core Subgroups, cosets and Lagrange's theorem Why the order of a subgroup divides the order of the group. divisors of 12 Core Cyclic groups and permutation groups Generators, orders, cycles, transpositions, parity. inverse of 5 mod 12 Core Homomorphisms, normal subgroups and quotient groups Structure-preserving maps, kernels, and the first isomorphism theorem. 12 mod 5 Core Rings and fields Two operations: integers, polynomials, ℤ/n, and when division is possible. inverse of 7 mod 11 Advanced Galois theory: why the quintic has no formula Field extensions, the Galois group, and solvability by radicals. x^3 + x + 1 = 0

Take the rules that integers obey, keep only the rules, and ask what else obeys them. Groups capture symmetry, rings capture arithmetic, fields capture solvability. Galois used them to prove the quintic has no formula.

更多经验教训

Preliminaries

A Short Note on ProofsSets and Equivalence RelationsSage

The Integers

Mathematical InductionThe Division AlgorithmSage

Groups

Integer Equivalence Classes and SymmetriesDefinitions and ExamplesSubgroupsSageGroup theoryGroup (mathematics)SubgroupCosetLagrange's theorem (group theory)Cyclic groupSymmetric groupPermutation groupNormal subgroupQuotient groupGroup homomorphismGroup actionSylow theoremsAbelian groupDihedral groupFree groupClassification of finite simple groups

Cyclic Groups

Cyclic SubgroupsMultiplicative Group of Complex NumbersThe Method of Repeated SquaresSage

Permutation Groups

Definitions and NotationDihedral GroupsSage

Cosets and Lagrange's Theorem

CosetsLagrange's TheoremFermat's and Euler's TheoremsSage

Introduction to Cryptography

Private Key CryptographyPublic Key CryptographySage

Algebraic Coding Theory

Error-Detecting and Correcting CodesLinear CodesParity-Check and Generator MatricesEfficient DecodingSage

Isomorphisms

Definition and ExamplesDirect ProductsSage

Normal Subgroups and Factor Groups

Factor Groups and Normal SubgroupsThe Simplicity of the Alternating GroupSage

Homomorphisms

Group HomomorphismsThe Isomorphism TheoremsSage

Matrix Groups and Symmetry

Matrix GroupsSymmetry

The Structure of Groups

Finite Abelian GroupsSolvable GroupsSage

Group Actions

Groups Acting on SetsThe Class EquationBurnside's Counting TheoremSage

The Sylow Theorems

The Sylow TheoremsExamples and ApplicationsSage

Rings

RingsIntegral Domains and FieldsRing Homomorphisms and IdealsMaximal and Prime IdealsAn Application to Software DesignSage

Polynomials

Polynomial RingsThe Division AlgorithmIrreducible PolynomialsSage

Integral Domains

Fields of FractionsFactorization in Integral DomainsSage

Lattices and Boolean Algebras

LatticesBoolean AlgebrasThe Algebra of Electrical CircuitsSage

Vector Spaces

Definitions and ExamplesSubspacesLinear IndependenceSage

Fields

Extension FieldsSplitting FieldsGeometric ConstructionsSage

Finite Fields

Structure of a Finite FieldPolynomial CodesSage

Galois Theory

Field AutomorphismsThe Fundamental TheoremApplicationsSage

Rings and fields

Ring (mathematics)Ideal (ring theory)Quotient ringIntegral domainPolynomial ringUnique factorization domainPrincipal ideal domainField (mathematics)Field extensionFinite fieldSplitting fieldAlgebraically closed fieldCharacteristic (algebra)

Galois and beyond

Galois groupSolvable groupAbel–Ruffini theoremModule (mathematics)Representation theoryLie algebraCategory of groupsUniversal algebra

其中一些经验教训从开放文本中改编: Judson, Abstract Algebra: Theory and Applications, Wikipedia. 每页课目都列出其来源和执照。

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人们对人们提出的问题

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

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