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Abstract Algebra

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.

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

Chapters from Judson, Abstract Algebra: Theory and Applications

Every section of the book, condensed into a lesson with its own practice problems.

1. Preliminaries

A Short Note on ProofsSets and Equivalence RelationsSagePreliminaries: exercises

2. The Integers

Mathematical InductionThe Division AlgorithmSageThe Integers: exercises

3. Groups

Integer Equivalence Classes and SymmetriesDefinitions and ExamplesSubgroupsSageGroups: exercises

4. Cyclic Groups

Cyclic SubgroupsMultiplicative Group of Complex NumbersThe Method of Repeated SquaresSageCyclic Groups: exercises

5. Permutation Groups

Definitions and NotationDihedral GroupsSagePermutation Groups: exercises

6. Cosets and Lagrange's Theorem

CosetsLagrange's TheoremFermat's and Euler's TheoremsSageCosets and Lagrange's Theorem: exercises

7. Introduction to Cryptography

Private Key CryptographyPublic Key CryptographySageIntroduction to Cryptography: exercises

8. Algebraic Coding Theory

Error-Detecting and Correcting CodesLinear CodesParity-Check and Generator MatricesEfficient DecodingSageAlgebraic Coding Theory: exercises

9. Isomorphisms

Definition and ExamplesDirect ProductsSageIsomorphisms: exercises

10. Normal Subgroups and Factor Groups

Factor Groups and Normal SubgroupsThe Simplicity of the Alternating GroupSageNormal Subgroups and Factor Groups: exercises

11. Homomorphisms

Group HomomorphismsThe Isomorphism TheoremsSageHomomorphisms: exercises

12. Matrix Groups and Symmetry

Matrix GroupsSymmetryMatrix Groups and Symmetry: exercises

13. The Structure of Groups

Finite Abelian GroupsSolvable GroupsSageThe Structure of Groups: exercises

14. Group Actions

Groups Acting on SetsThe Class EquationBurnside's Counting TheoremSageGroup Actions: exercises

15. The Sylow Theorems

The Sylow TheoremsExamples and ApplicationsSageThe Sylow Theorems: exercises

16. Rings

RingsIntegral Domains and FieldsRing Homomorphisms and IdealsMaximal and Prime IdealsAn Application to Software DesignSageRings: exercises

17. Polynomials

Polynomial RingsThe Division AlgorithmIrreducible PolynomialsSagePolynomials: exercises

18. Integral Domains

Fields of FractionsFactorization in Integral DomainsSageIntegral Domains: exercises

19. Lattices and Boolean Algebras

LatticesBoolean AlgebrasThe Algebra of Electrical CircuitsSageLattices and Boolean Algebras: exercises

20. Vector Spaces

Definitions and ExamplesSubspacesLinear IndependenceSageVector Spaces: exercises

21. Fields

Extension FieldsSplitting FieldsGeometric ConstructionsSageFields: exercises

22. Finite Fields

Structure of a Finite FieldPolynomial CodesSageFinite Fields: exercises

23. Galois Theory

Field AutomorphismsThe Fundamental TheoremApplicationsSageGalois Theory: exercises

Symbols used here

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
(G, \cdot),\ e,\ g^{-1}
group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.
G \cong H,\ G / N
isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.
\operatorname{Hom}(A, B),\ f \circ g
arrows from A to B, composition
The set of morphisms; do g then f.

Questions people ask

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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