Part V: Algebraic Structures

Algebra is the study of sets equipped with operations satisfying specified axioms. The key insight of modern algebra (Emmy Noether, van der Waerden) is that many seemingly different structures share the same abstract form, and theorems about the form apply uniformly to all of them.

5.1 The Tower of Group-like Structures

Structure	Operations	Axioms added (cumulatively)
Magma	·	Binary operation (closure)
Semigroup	·	Associativity
Monoid	·	Identity element
Group	·	Inverses for all elements
Abelian group	+	Commutativity

Group: A set G with an associative binary operation, an identity element e, and inverses for every element. Groups are the mathematical model of symmetry. Every geometric transformation, every permutation, every change of coordinates is (or lives in) a group.

Key theorems:

• Lagrange's theorem: If H is a subgroup of finite group G, then |H| divides |G|
• Cayley's theorem: Every group is isomorphic to a subgroup of a symmetric group
• Sylow theorems: For a prime p dividing |G|, G contains subgroups of order pⁿ for each power pⁿ | |G|
• Jordan-Hölder theorem: Every finite group has a unique composition series (up to order and isomorphism of factors)

Classification of finite simple groups (CFSG): The finite simple groups — the prime atoms of group theory — are exactly:

1. Cyclic groups ℤ_p (p prime)
2. Alternating groups A_n (n ≥ 5)
3. Groups of Lie type (infinite families)
4. The 26 sporadic groups, culminating in the Monster group of order ≈ 8×10⁵³

The proof spans ~15,000 journal pages and is the longest proof in mathematics.

5.2 Rings

Definition: A ring (R, +, ×) has two binary operations where (R,+) is an abelian group, (R,×) is associative with identity 1, and multiplication distributes over addition.

Type	Additional axiom
Commutative ring	xy = yx for all x, y
Integral domain	No zero divisors: xy = 0 ⟹ x=0 or y=0
Principal ideal domain (PID)	Every ideal is principal (generated by one element)
Unique factorization domain (UFD)	Unique factorization into irreducibles
Euclidean domain	Division algorithm exists
Field	Every non-zero element has a multiplicative inverse

The hierarchy: Field ⊂ Euclidean domain ⊂ PID ⊂ UFD ⊂ Integral domain ⊂ Commutative ring ⊂ Ring.

Examples:

• ℤ is a PID but not Euclidean (well, actually ℤ is Euclidean with the norm n ↦ |n|)
• ℤ[√−5] is a UFD failure: 6 = 2×3 = (1+√−5)(1−√−5)
• Every field is a Euclidean domain trivially
• k[x] (polynomials over a field) is a Euclidean domain with degree as the norm

Ideals and quotients: An ideal I ⊆ R is a subgroup closed under multiplication by any ring element. The quotient ring R/I has elements [r] = r + I, with [r][s] = [rs]. The ideal structure of a ring determines much of its arithmetic.

5.3 Fields

Extensions: If k ⊆ K are fields, K is a field extension of k. The degree [K:k] is the dimension of K as a k-vector space.

Algebraic elements: α ∈ K is algebraic over k if it satisfies a polynomial with k-coefficients. The minimal polynomial is the unique monic irreducible polynomial of lowest degree satisfied by α.

Algebraic closure: Every field k has an algebraic closure k̄ — the smallest algebraically closed field containing k. ℂ is the algebraic closure of ℝ.

Galois theory: The Galois group Gal(K/k) is the group of field automorphisms of K fixing k pointwise. The fundamental theorem of Galois theory establishes a bijection between subfields of K containing k and subgroups of Gal(K/k), reversing inclusion.

Application: A polynomial is solvable by radicals if and only if its Galois group is a solvable group. The general quintic equation has Galois group S₅, which is not solvable; therefore, there is no general quintic formula in radicals. (Abel-Ruffini theorem, Galois 1832.)

5.4 Vector Spaces and Linear Algebra

Definition: A vector space over field k is an abelian group (V,+) with a scalar multiplication k×V→V satisfying the eight vector space axioms.

Key structures:

• Basis: A linearly independent spanning set. Every vector space has a basis (requires Axiom of Choice); any two bases of the same vector space have the same cardinality (the dimension)
• Linear map (homomorphism): A function T: V→W preserving addition and scalar multiplication
• Dual space: V* = Hom_k(V, k) — the vector space of linear functionals on V
• Eigenvalue / eigenvector: Tv = λv for scalar λ and non-zero vector v

Fundamental theorem of linear algebra: For T: V→W a linear map between finite-dimensional spaces:

• dim(ker T) + dim(im T) = dim V
• im T = (ker T*)⊥

Spectral theorem: For a symmetric (self-adjoint) operator on a finite-dimensional inner product space, there exists an orthonormal basis of eigenvectors with real eigenvalues. Generalization: any normal operator on a finite-dimensional complex inner product space is diagonalizable.

5.5 Modules

Definition: A module over ring R is an abelian group M with scalar multiplication R×M→M satisfying the module axioms. (Vector spaces are modules over fields; modules relax the field condition to a ring.)

Free modules: Analogous to vector spaces; have a basis, and any element is a unique R-linear combination of basis elements.

Projective and injective modules: Projective modules generalize free modules; injective modules are the "dual" notion. These are homological algebra concepts characterizing how exact sequences split.

Tensor products: The tensor product M ⊗_R N of R-modules captures bilinear maps from M×N. Fundamental in multilinear algebra, differential geometry, and quantum mechanics.

5.6 Lattices and Boolean Algebras

Lattice: A partially ordered set where every pair of elements has a least upper bound (join, ∨) and greatest lower bound (meet, ∧).

Distributive lattice: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c).

Boolean algebra: A distributive lattice with complement. Every element a has a complement aᶜ with a ∨ aᶜ = 1 and a ∧ aᶜ = 0.

Stone's Representation Theorem: Every Boolean algebra is isomorphic to the algebra of clopen (simultaneously open and closed) subsets of a compact, Hausdorff, totally disconnected topological space (a Stone space). This establishes a duality between Boolean algebras and Stone spaces.

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