Algebraic structures: groups, rings, fields, modules, vector spaces and their morphisms. Substrate for linear algebra (matrices, eigenvalues, tensors), field theory (Galois), and the algebraic side of quantum mechanics and machine learning.
abstract-algebra
Group (G, ·)
A set G with a binary operation · satisfying closure, associativity, identity, and inverses. Abelian if · is commutative. Formalises…
Subgroup
A subset H ⊆ G that is itself a group under G's operation. Lagrange: |H| divides |G| when G is finite.
Group homomorphism
A map φ: G → H with φ(ab) = φ(a)φ(b). Kernel = φ⁻¹(e_H) is a normal subgroup of G; image is a subgroup of H. First isomorphism theorem:…
Ring (R, +, ·)
A set with two operations: (R, +) is an abelian group, · is associative with identity, and · distributes over +. Examples: ℤ, polynomial…
Ideal
A subset I ⊆ R closed under + and absorbing · (rI ⊆ I for all r). Quotient R/I is a ring. Maximal ideals correspond to points of the prime…
Field
A commutative ring in which every nonzero element has a multiplicative inverse. Examples: ℚ, ℝ, ℂ, 𝔽_p for p prime. The algebraic backbone…
Polynomial ring k[x]
The ring of polynomials with coefficients in a field k. k[x] is a Euclidean domain; irreducible polynomials play the role of primes.
Vector space
A set V with addition + and scalar multiplication over a field k, satisfying the eight vector-space axioms. Has a basis; dimension is an…
Linear map
A map T: V → W with T(au + bv) = aT(u) + bT(v). In finite dimensions, a linear map is represented by a matrix once bases are chosen.
Matrix and determinant
An n×n matrix is a linear operator on kⁿ once a basis is fixed. Determinant det(A) = Σ_σ sgn(σ) ∏ a_{i,σ(i)} detects invertibility (A…
Eigenvalue / eigenvector
For A: V → V, a scalar λ and nonzero v with Av = λv. The characteristic polynomial det(A − λI) = 0 gives spectrum. Basis of spectral…
Tensor product V ⊗ W
The universal bilinear construction: bilinear maps V × W → X are in bijection with linear maps V ⊗ W → X. Substrate for multilinear…
Galois theory
Correspondence between subfields of a field extension K/F and subgroups of its automorphism group Gal(K/F). Classical corollary: quintic…
Fundamental theorem of algebra
Every non-constant polynomial in ℂ[x] has a complex root. Equivalent statement: ℂ is algebraically closed. Proofs use complex analysis or…
Boolean algebra
A complemented distributive lattice on a two-element set {0, 1} with meet (∧), join (∨), and complement (¬). Equivalent (as an algebraic…
δ_S (silver ratio)
The positive root of x² = 2x + 1 — i.e. 1 + √2. Plays the analogue of φ for the Pell sequence; appears in octagonal symmetry.
ψ (supergolden ratio)
The unique real root of x³ = x² + 1. The growth rate of the Narayana-cow sequence, and the cubic cousin of φ.
ψ (reciprocal-Fibonacci constant)
Sum of the reciprocals of the Fibonacci numbers. Proven irrational by André-Jeannin (1989); whether transcendental is open.
Normal subgroup
A subgroup N ≤ G with gNg^{-1} = N for all g ∈ G (equivalently, N is the kernel of a homomorphism). Enables quotient groups.
Quotient group
G/N := {gN : g ∈ G} with (gN)(hN) = (gh)N. Well-defined iff N is normal; universal among homomorphisms from G killing N.
Isomorphism theorems (groups/rings/modules)
First: G/ker φ ≅ im φ for any homomorphism φ. Second: (HN)/N ≅ H/(H∩N) when N normal. Third: (G/N)/(K/N) ≅ G/K when N ⊆ K, both normal. …
Sylow theorems
For |G|=p^a m (p∤m): (I) Sylow p-subgroups of order p^a exist; (II) they are all conjugate; (III) their number n_p ≡ 1 (mod p) and divides…
Solvable group
A group G admitting a finite subnormal series 1 = G_0 ◁ G_1 ◁ … ◁ G_n = G with each G_{i+1}/G_i abelian. Galois: an equation is solvable…
Simple group
A non-trivial group with no non-trivial proper normal subgroups. Classification of finite simple groups (2004) identifies them as cyclic…
Free group
The group F_S on a set S consists of reduced words in S and S^{-1}; it satisfies the universal property that any function S → G (group)…
Symmetric group S_n
The group of all bijections of {1,…,n} under composition, order n!. Cayley: every finite group embeds in some S_n. A_n is simple for n ≥…
Lie algebra
A vector space 𝔤 with a bilinear bracket [·,·] that is anti-symmetric and satisfies the Jacobi identity. Tangent structure of Lie groups;…
Representation theory
Studies linear actions of algebraic structures (groups, rings, Lie algebras, associative algebras) on vector spaces; reduces structure to…
Schur's lemma
A G-equivariant map between irreducible representations is either zero or an isomorphism; over an algebraically-closed field it is a…
UFD / PID / Euclidean domain hierarchy
Integral domains satisfy strict inclusions Euclidean ⊂ PID ⊂ UFD ⊂ (ID), with e.g. ℤ[(1+√−19)/2] a PID but not Euclidean, and ℤ[x] a UFD…
Noetherian ring
A ring in which every ascending chain of ideals stabilises (equivalently, every ideal is finitely generated). Hilbert basis theorem:…
Module over a ring
For a ring R, an R-module M is an abelian group with an R-action R×M→M satisfying distributivity and (rs)m = r(sm). Generalises vector…
Exterior algebra Λ(V)
The quotient T(V)/⟨v⊗v⟩ of the tensor algebra; the universal associative algebra in which v² = 0. Graded components Λ^k V encode…
Lie group
A smooth manifold G that is also a group such that multiplication (x, y) ↦ xy and inversion x ↦ x⁻¹ are smooth maps. Canonical examples:…
Jacobi identity
For a Lie algebra with bracket [·,·], [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0 for all X, Y, Z. Replaces the associative law (which…
Nakayama's lemma
Finitely-generated M over (R,m) local: mM = M ⟹ M = 0. Equiv: M/mM generates M via any lift. Workhorse of commutative algebra.
Krull dimension
dim R = sup length of chain p_0 ⊊ p_1 ⊊ … ⊊ p_n of prime ideals. Catches geometric dimension of Spec R; Noether normalization.
Noether normalization
Finitely-generated k-algebra A of dim d: exists algebraically independent y_1,…,y_d with A finite over k[y_1,…,y_d]. Projects varieties to…
Structure theorem over PID
Finitely-generated module M over PID: M ≅ R^r ⊕ R/(d_1) ⊕ … ⊕ R/(d_s) with d_1 | d_2 | … | d_s. Specializes to Jordan form, invariant…
Cartan-Killing classification
Complex simple Lie algebras = A_n, B_n, C_n, D_n, G_2, F_4, E_6, E_7, E_8. Classified by Dynkin diagrams; root systems. Killing 1890,…
Grothendieck K-theory K_0(R)
Abelian group on iso classes of f.g. projective R-modules modulo [P⊕Q] = [P]+[Q]. K_0 of a ring; higher K-theory (Quillen) detects…
Artinian vs Noetherian
Noetherian: ACC on ideals; Artinian: DCC on ideals. Hopkins-Levitzki: Artinian ⟹ Noetherian (rings). Length, composition series.
Dedekind domain
Integrally-closed Noetherian domain with every nonzero prime maximal. Ideals factor uniquely into primes. Ring of integers O_K; class group…
UFD ⊃ PID ⊃ Euclidean domain
Euclidean ⟹ PID ⟹ UFD ⟹ integral domain. Each inclusion strict: ℤ[(1+√−19)/2] PID not Euclidean; ℤ[x] UFD not PID.
Galois descent
Recovers object over k from L-object + Gal(L/k)-action (twisted form). Forms of quadratic forms, Brauer groups, Galois cohomology H¹.
Brauer group Br(k)
Central simple k-algebras modulo matrix algebras under ⊗. Br(ℝ) = ℤ/2, Br(ℚ_p) = ℚ/ℤ, Br(F_q) = 0. Arises as H²(Gal, k̄*).
Hopf algebra
Bialgebra with antipode S. k[G] group, U(g) enveloping, O(G) functions. Quantum groups U_q(g) deform U(g) losing cocommutativity.