Alphabeta Math

Abstract Algebra

78 pages in 8 parts

Abstract algebra develops structures defined by operations and the maps that preserve them. Its group sequence moves from subgroups, cosets, quotients and homomorphisms to actions, products, presentations, finite-group structure and solvability. Ring topics include ideals, quotient rings, divisibility, Euclidean and principal ideal domains, polynomial rings and localization. Field theory covers algebraic extensions, finite fields, splitting fields, separability, algebraic closures, cyclotomic extensions and Galois correspondence. Modules and exact sequences lead to tensor products, chain conditions, semisimplicity and canonical forms; the collection also develops finite-group characters and induced representations.

Some topics draw on Linear Algebra, Lie Theory and Functional Analysis, while others use results from Number Theory and Real Analysis. Algebraic structures and representation methods from this collection serve later work in Commutative Algebra, Algebraic Geometry, Representation Theory of Groups, Symmetric Groups and Symmetric Functions, and Coxeter Groups. Their results also recur in Combinatorics, Complex Analysis, Topology, Quantum Computing and related subjects, where groups, rings, fields and modules provide essential algebraic tools.

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

  1. Part 1 · Groups, quotients and homomorphisms

    5 pages

    The naturals, the integers, the rationals and the reals were each built with their own proofs of the same handful of facts, and a group is what those facts have in common: one associative operation, an identity, inverses. Cosets partition a group and give Lagrange's theorem, normality is the condition under which the cosets themselves form a group, and the isomorphism theorems say that every homomorphism is a quotient followed by an inclusion.

  2. Part 2 · Actions, permutations and presentations

    3 pages · after Part 1

    A group acts when its elements become symmetries of a set, and the orbit-stabiliser relation turns a question about the group into a count. Cayley's theorem embeds any group in a symmetric group, cycle decomposition and the sign homomorphism describe permutations, and the alternating group is simple for degree at least five. In the other direction a free group has no relations at all, so every group is a quotient of one, which is what a presentation and an amalgamated free product are built from.

  3. Part 3 · Finite and solvable groups

    2 pages · after Parts 1 and 2

    A finite abelian group is a direct product of cyclic groups of prime power order, and the decomposition is unique, so the classification of that class is complete. A composition series breaks any finite group into simple pieces, Jordan-Holder says the pieces do not depend on the series, and solvability is the case where each piece is abelian.

  4. Part 4 · Rings, ideals and polynomials

    5 pages · after Parts 1 and 3

    A ring has two operations, and the interaction between them is what the earlier number systems kept re-proving. Ideals are the subobjects a quotient can be taken by, and the same isomorphism theorems hold. Divisibility then generalises: a Euclidean domain has division with remainder, every Euclidean domain is a principal ideal domain, and every principal ideal domain has unique factorisation. Polynomial rings supply the working examples, and the field of fractions and localisation invert what a domain lacks.

  5. Part 5 · Field extensions

    5 pages · after Part 4

    Adjoining a root of an irreducible polynomial is a quotient of a polynomial ring by a maximal ideal, which is how the complex numbers appear here rather than by decree. A splitting field factors a polynomial completely and is unique up to isomorphism; degrees multiply in towers, which classifies the finite fields; symmetric polynomials express what depends on the roots but not their order; and an algebraic closure exists, with embeddings and separability measuring how many extensions of a map there are.

  6. Part 6 · Modules

    5 pages · after Parts 2, 4 and 5

    A module is a vector space over a ring instead of a field, so a submodule need not be a direct summand and dimension need not exist. Exact sequences state that failure, free, projective and injective modules describe when it can be repaired, and the tensor product linearises bilinear maps. Chain conditions bound it from the other side, and Wedderburn-Artin says which rings have none. Over a principal ideal domain it disappears: a submodule of a finite free module has an aligned basis, yielding invariant factors, elementary divisors, Smith normal form and their uniqueness. Over the integers that classifies the finitely generated abelian groups; over a polynomial ring it gives rational canonical form, Cayley-Hamilton, Jordan form, and the similarity criterion.

  7. Part 7 · Sylow theory and split extensions

    11 pages · after Parts 2, 3, 4, 5 and 6

    A split extension is a semidirect product, so actions assemble groups from normal pieces, and Sylow theory turns the prime divisors of ∣G∣ into existence, conjugacy, and counting statements for p-subgroups. Nilpotence, the Frattini subgroup, the Galois correspondence, finite fields, cyclotomic extensions, solvability by radicals, and Artin's proof of the fundamental theorem of algebra show how extensions are built. Representation theory then passes to k[G]-modules, where Maschke gives semisimplicity. Finite averaging supplies the intervening linear-algebra and character-theory interface: projections, idempotent traces, class-function indicators, the normalized Hermitian form, equality for finite unit sums, and the Galois behavior of cyclotomic averages. Characters can then encode orthogonality and central-character arithmetic, while induction adds Frobenius reciprocity, Mackey decomposition, and Burnside's paqb solvability theorem.

  8. Part 8 · Algebra and representation interfaces

    3 pages · after Parts 5 and 6

    These draft extensions connect the existing foundations to algebraic structures and representation problems motivated by theoretical physics. Each companion supplies the planned examples and boundary cases; exact hypotheses, local proof obligations, and reuse holds remain in its page contract.