Algebra.

OpeningWhat algebra is.

Algebra is the mathematics of structure: of operations, of equations, of the symmetries that connect one quantity to another. It begins with the idea that a number you don't yet know can still be reasoned about — that x is a respectable noun.

The word comes from the Arabic al-jabr ("the restoration"), the title verb of Muhammad ibn Musa al-Khwarizmi's c. 820 treatise on solving linear and quadratic equations. The discipline now spans school exercises in linear systems, the abstract architecture of groups and rings, and the machinery underneath modern physics and machine learning.

This deck traces the line from Babylonian tablets to the Langlands program — thirty-odd leaves on the most generative branch of mathematics.

Chapter IBefore the symbol.

The Babylonians (c. 1800 BCE) solved quadratic equations by recipe: tablet YBC 6967 prescribes a procedure equivalent to completing the square, with no symbolic notation. The Egyptians worked linear equations in the Rhind papyrus (c. 1650 BCE) using the method of false position.

The Greeks generally preferred geometry to algebra; Euclid's Book II proves identities like (a+b)² = a² + 2ab + b² as statements about rectangles. The exception is Diophantus of Alexandria (c. 250 CE), whose Arithmetica introduced syncopated abbreviations for the unknown and its powers and gave an entire book to what we now call Diophantine equations — integer solutions to polynomial equations.

Diophantus's text was lost to Western Europe for a millennium. When it resurfaced in the Renaissance, it set off the modern subject.

Chapter IIThe word itself.

Muhammad ibn Musa al-Khwarizmi, working at Baghdad's House of Wisdom under Caliph al-Ma'mun, completed al-Kitab al-Mukhtasar fi Hisab al-Jabr wa'l-Muqabala ("The Compendious Book on Calculation by Restoration and Balancing") around 820 CE.

The book classified linear and quadratic equations into six standard forms and gave geometric demonstrations for solving each. Crucially, it treated the unknown as an object you could manipulate by rules — adding equal quantities to both sides ("restoration"), cancelling like terms ("balancing").

The Latin transliteration algoritmi, from his name, gave us "algorithm." The Arabic al-jabr gave us "algebra." The system of place-value Hindu-Arabic numerals he popularised in another treatise gave Europe arithmetic.

Chapter IIICardano, 1545.

Gerolamo Cardano's Ars Magna (1545) published the first general solutions to the cubic and quartic equations — a Renaissance breakthrough that broke a 3,000-year impasse.

The cubic formula was actually discovered by Scipione del Ferro (c. 1515) and rediscovered by Niccolò Tartaglia (1535). Cardano extracted Tartaglia's method under oath of secrecy, then published it after finding del Ferro's earlier work — provoking a feud that ended in a public mathematical duel between Tartaglia and Cardano's student Lodovico Ferrari (who himself solved the quartic).

The cubic forced an unwelcome guest into mathematics: imaginary numbers. To find real roots of certain cubics, the formula demanded square roots of negatives in intermediate steps. Cardano called these "as subtle as it is useless." Rafael Bombelli (1572) took them seriously and built the first coherent rules for complex arithmetic.

Chapter IVSymbolic notation.

François Viète's In artem analyticem isagoge (1591) made the decisive break: use letters for both knowns and unknowns. Viète used vowels for unknowns, consonants for parameters. For the first time you could write an equation that stood for a class of problems, not a single problem.

Thomas Harriot (posthumous, 1631) introduced the modern signs > and < and lowered the visual weight of algebraic expression. René Descartes's La Géométrie (1637) gave us today's convention: letters near the start of the alphabet (a, b, c) for parameters; letters near the end (x, y, z) for unknowns; and exponential notation x², x³.

The symbolic revolution mattered as much as the analytic ones. Once you can write ax² + bx + c = 0, you can think about the structure of the solution rather than re-derive it for every triple of numbers.

Chapter VAlgebraic geometry, version one.

Descartes's La Géométrie (1637), an appendix to his Discourse on the Method, fused algebra with geometry. A point in the plane became an ordered pair (x, y); a curve became the locus of solutions to an equation.

This single move — coordinates — let geometric questions be answered by algebraic manipulation, and let algebraic relationships be visualised. Pierre de Fermat developed analytic geometry independently and slightly earlier, but his work circulated in manuscript and Descartes's printed text won the field.

The unification was decisive for what would come. Calculus, complex analysis, differential geometry, and modern algebraic geometry all rest on the Cartesian premise that shape and equation are the same kind of thing.

Chapter VIVectors and matrices.

The mathematics of arrows and arrays. A vector is an element of a vector space — an object that can be added to another vector and scaled by a number. The familiar (x, y, z) in 3-space is the canonical example; the bytes of a JPEG, the activations of a neural-network layer, and a polynomial of bounded degree are also vectors in their respective spaces.

A matrix is a rectangular array of numbers that encodes a linear transformation between vector spaces. A·v = w sends the vector v to the vector w. Matrix multiplication composes transformations.

The discipline crystallised in the 19th century with Arthur Cayley's 1858 memoir on matrices and Hermann Grassmann's 1844 Ausdehnungslehre on the abstract algebra of vector spaces. It is now the dominant computational substrate of applied mathematics — graphics, physics simulation, statistics, and machine learning all run on linear algebra.

Chapter VIISolving for many unknowns.

A system of linear equations in n unknowns has the form A·x = b, where A is an m×n matrix. Three possibilities: a unique solution, infinitely many solutions, or no solution at all.

Gaussian elimination (Carl Friedrich Gauss formalised it c. 1810; the underlying procedure was known to the Chinese Nine Chapters, c. 200 BCE) reduces the system to row-echelon form by elementary row operations. LU decomposition, QR decomposition, and singular value decomposition are the modern numerical-linear-algebra factorisations descendant from it.

The complexity of solving an n×n dense linear system is O(n³) by elimination, reduced to O(n^2.371) by current fast matrix-multiplication algorithms — a theoretical curiosity, since the constants make these slower than Gaussian elimination at any practical size.

Chapter VIIIThe single number.

Every square matrix has a determinant — a single scalar that captures whether the matrix is invertible (nonzero) or singular (zero). Geometrically, the determinant is the signed volume of the parallelepiped spanned by the matrix's columns.

The 2×2 case: det([a b; c d]) = ad − bc. The 3×3 case expands as a sum of six terms with alternating signs. The general n×n case is the Leibniz formula — a sum over the n! permutations of n elements, weighted by their signs.

Determinants gave the first solution method for general linear systems via Cramer's rule (1750). They are now mostly conceptual scaffolding rather than computational tools — for an n×n system, computing the determinant directly is wildly inefficient compared to Gaussian elimination. But the concept underlies orientation, change-of-variables in integration, and the Jacobian of a map.

Chapter IXVectors that don't turn.

An eigenvector of a matrix A is a nonzero vector v such that A·v = λ·v for some scalar λ — the corresponding eigenvalue. Geometrically: v points in a direction the transformation merely stretches (by λ) rather than rotates.

The eigenvalues of A are the roots of its characteristic polynomial det(A − λI) = 0. The fundamental theorem of algebra then guarantees an n×n matrix has n eigenvalues over the complex numbers (counting multiplicity).

Eigenvalues control nearly every dynamic behaviour of a linear system: the long-term growth rate of A^k v, the stability of differential equations, the principal axes of an ellipsoid. Principal component analysis, Google's PageRank, vibration analysis, and quantum mechanics all reduce to eigenproblems.

Chapter XThe abstract framework.

A vector space over a field F is a set V with two operations — addition v + w and scalar multiplication α·v — satisfying eight axioms (associativity, commutativity, identity, inverses for addition; compatibility, distributivity, identity for scalar multiplication).

The abstraction is liberating. The same theorems apply to ordered n-tuples of real numbers, polynomials of degree at most n, continuous functions on an interval, and the bytes of a digital image. A basis is a linearly-independent spanning set; the dimension is the cardinality of any basis.

Most of linear algebra is the study of linear transformations between vector spaces and the structures (subspaces, kernels, images, quotients) they create. The framework was solidified by Hermann Weyl, Stefan Banach, and Emmy Noether in the 1920s.

Chapter XIThe unsolvable quintic.

By 1700 mathematicians had general formulas — in radicals — for polynomials of degrees 2, 3, and 4. The quintic resisted. Paolo Ruffini (1799, with gaps) and Niels Henrik Abel (1824, complete) proved that no general formula in radicals exists for degree 5 or higher. The Abel–Ruffini theorem.

Évariste Galois (1811–1832) gave the deeper answer. The night before his fatal duel — at age 20 — he wrote a letter to Auguste Chevalier sketching a theory that associates to each polynomial a group of permutations of its roots. The polynomial is solvable in radicals exactly when this group is solvable in a precise group-theoretic sense.

The quintic's symmetric group S₅ is not solvable. Hence the impossibility. Galois's theory transformed algebra: the central object of study was no longer the equation but the symmetry group attached to it.

Chapter XIIThe mathematics of symmetry.

A group is a set G with a binary operation satisfying four axioms: closure, associativity, identity, inverses. The motivating example is symmetries of an object — rotations of a cube, permutations of a set, the integers under addition.

The concept generalises by stripping geometry away. Anything that "transforms reversibly" forms a group. Two crystals with the same symmetry group are mathematically interchangeable for many physical purposes; the standard model of particle physics is built from gauge groups; cryptographic protocols rely on the discrete-logarithm problem in finite groups.

The classification of all finite simple groups, completed in 2004 after a 50-year collaborative effort spanning hundreds of papers and 10,000+ pages, identifies exactly 18 infinite families plus 26 sporadic groups. The largest sporadic, the Monster (~8×10⁵³ elements), connects unexpectedly to modular forms via "monstrous moonshine" — proven by Borcherds in 1992.

Chapter XIIIThe four laws.

Pinned down: (G, ·) is a group if

If the operation is also commutative — a · b = b · a — the group is called abelian, after Niels Henrik Abel. The integers under addition, the nonzero reals under multiplication, and the additive group of any vector space are abelian. Most interesting symmetry groups are not.

From these four lines, an enormous body of theorems follows. Lagrange's theorem (the order of a subgroup divides the order of the group), Sylow's theorems on p-subgroups, the structure theorem for finitely generated abelian groups — all from four axioms.

Chapter XIVTwo essential families.

The cyclic group Z/nZ: integers modulo n, addition. The simplest infinite-family of finite groups. A cyclic group has a single generator; every element is g, g², g³, … for some g. Models clock arithmetic, the rotational symmetry of a regular n-gon.

The symmetric group S_n: all permutations of n objects, with n! elements. S₃ has 6 elements (the symmetries of a triangle, rotations and reflections combined); S₄ has 24 (proper symmetries of a tetrahedron, plus mirror images); S₅ has 120 — and is non-abelian, non-solvable, the structural reason the quintic has no radical formula.

Cayley's theorem (1854): every group is isomorphic to a subgroup of some symmetric group. The symmetric groups are universal — every conceivable abstract group lives inside one.

Chapter XVTwo operations.

A ring is a set with two operations — addition (always abelian-group-like) and multiplication (associative, distributing over addition). The integers Z are the canonical ring. So are polynomials with real coefficients, square matrices of fixed size, continuous functions on an interval.

Multiplication need not be commutative (matrices), need not have an identity (some authors), need not have inverses (the integers have no multiplicative inverse for 2). The variations carve up the subject: commutative rings, integral domains, principal ideal domains, unique factorisation domains.

Emmy Noether's 1921 paper Idealtheorie in Ringbereichen established the modern abstract theory of ideals — subsets closed under multiplication by the whole ring. The Noetherian condition (every ideal finitely generated) is the foundation of commutative algebra and algebraic geometry.

Chapter XVIWhere division works.

A field is a commutative ring in which every nonzero element has a multiplicative inverse. The rationals Q, the reals R, the complex numbers C, and the field of p-adic numbers are the canonical infinite fields.

The finite fields are stranger and more useful. For each prime power p^n there is exactly one field with that many elements (up to isomorphism), denoted F_{p^n} or GF(p^n). Évariste Galois introduced them in his 1830 paper on number theory; they are now the substrate of error-correcting codes (Reed–Solomon), elliptic-curve cryptography, and combinatorial design.

Field extensions — bigger fields containing a given one — are the technical heart of Galois theory. The Galois group of an extension is the group of field automorphisms fixing the smaller field; this is the symmetry group whose solvability decides whether a polynomial is solvable in radicals.

Chapter XVIIThe fundamental theorem.

Every non-constant polynomial with complex coefficients has a complex root. Equivalently: a polynomial of degree n has exactly n complex roots, counting multiplicity. The fundamental theorem of algebra.

The statement is older than its proofs. D'Alembert (1746) gave an incomplete proof; Gauss gave four proofs over his lifetime, the first in his 1799 doctoral thesis (also incomplete by modern standards), the last in 1849.

The theorem is, despite the name, not provable by purely algebraic means — every proof requires some topology or analysis. It says that the field C is algebraically closed: there is no need for further extensions to find roots. Compare R, where x² + 1 = 0 has no solution.

Chapter XVIIILinear algebra everywhere.

The applied dominance of linear algebra is hard to overstate. Computer graphics: every 3D transformation is a 4×4 matrix multiplication; the rasterisation pipeline is sequential matrix-vector products. Robotics: forward and inverse kinematics are matrix products and pseudo-inverses.

Machine learning: a feedforward neural network is a sequence of matrix multiplications interleaved with elementwise nonlinearities. Training is gradient descent on parameters. Modern GPUs are matrix-multiplication accelerators; nearly every recent breakthrough — image generation, large language models, protein structure — sits on top of optimised A·B.

Statistics: regression is least-squares, the projection of a vector onto the column space of a matrix. Quantum mechanics: states are vectors in a Hilbert space; observables are Hermitian operators (self-adjoint matrices in finite dimensions). Network analysis: the adjacency matrix and its eigendecomposition expose graph structure.

Chapter XIXThe ring R[x].

Polynomial equations were algebra's first object. They remain central. The polynomial ring R[x] over a ring R consists of finite formal sums a₀ + a₁x + a₂x² + …; the multivariate ring R[x₁,…,x_n] generalises.

For polynomials over a field, division algorithms work: there is a polynomial GCD (the Euclidean algorithm), unique factorisation into irreducibles, and explicit root-finding methods over C. Multivariate polynomials introduce Gröbner bases (Buchberger 1965) — the canonical computational tool for solving polynomial systems.

Polynomials are the workhorse interpolation, approximation, and coding-theory tool. Reed–Solomon codes treat data blocks as polynomial coefficients; Lagrange interpolation reconstructs them from any sufficient subset of evaluations. Most of practical computer algebra is polynomial manipulation.

Chapter XXBoolean algebra.

George Boole's An Investigation of the Laws of Thought (1854) algebraised logic. A Boolean algebra has elements 0 and 1 with operations AND, OR, NOT satisfying laws like x AND x = x (idempotence) — distinct from ordinary arithmetic.

The subject lay relatively unused until Claude Shannon's 1937 master's thesis at MIT, A Symbolic Analysis of Relay and Switching Circuits, observed that Boolean algebra is the right mathematics for designing electrical switching networks. Every digital circuit ever built is, at the logical level, a Boolean expression.

The connection runs deep. Stone's representation theorem (1936) shows every Boolean algebra is isomorphic to a field of subsets. The proposition calculus, set theory, and circuit logic are three faces of one structure.

Chapter XXIContinuous symmetry.

Sophus Lie (1842–1899) developed the theory of continuous transformation groups — now Lie groups — to do for differential equations what Galois had done for algebraic ones. A Lie group is a group that is also a smooth manifold; group operations and inverses are smooth.

The classical Lie groups: GL(n), SL(n), O(n), SO(n), U(n), SU(n), plus the symplectic groups. Each has an associated Lie algebra — its tangent space at the identity, with a bracket operation [X, Y] = XY − YX that captures infinitesimal group structure.

Lie groups are the language of symmetry in modern physics. Rotations: SO(3). Special relativity: the Lorentz group. Standard model gauge symmetry: SU(3) × SU(2) × U(1). Élie Cartan's 1894 thesis classified the simple Lie algebras over C — four infinite families and five exceptional cases.

Chapter XXIICategory theory.

A category is a collection of objects and arrows between them, with associative composition and identities. Samuel Eilenberg and Saunders Mac Lane introduced the framework in 1945 to make precise what "natural" means for transformations between mathematical structures.

The shift in perspective is profound: study mathematical objects through their relationships rather than their internal composition. Two categories can be related by a functor; two functors by a natural transformation. Most modern mathematics — algebraic topology, algebraic geometry, homological algebra — is fundamentally categorical.

Alexander Grothendieck's 1960s reconstruction of algebraic geometry on the foundation of schemes and topoi could not have been done in any other language. Functional programming languages descend from category theory by way of typed lambda calculus; monads in Haskell are literally the categorical concept.

Chapter XXIIIAlgebra in physics.

The 20th century turned physics into applied algebra. Werner Heisenberg's 1925 matrix mechanics formulated quantum mechanics as non-commuting operators on a Hilbert space. Paul Dirac's notation made the linear algebra explicit.

Symmetry principles drive everything. Emmy Noether's 1918 theorem: every continuous symmetry of an action gives a conservation law. Translation invariance gives momentum conservation, rotational invariance gives angular momentum, time-translation invariance gives energy.

The standard model is built from the gauge group SU(3) × SU(2) × U(1); particles are representations of this group. Eugene Wigner's 1939 paper classified the irreducible representations of the Poincaré group — these are the relativistic elementary particles. Algebra isn't applied to particle physics; it is particle physics.

Chapter XXIVEquations as spaces.

The study of geometric objects defined by polynomial equations. An algebraic variety is the zero locus of a system of polynomials. x² + y² = 1 is the unit circle; y² = x³ + ax + b is an elliptic curve.

The classical (19th-century) subject — Riemann surfaces, projective curves, the Italian school of Castelnuovo, Enriques, Severi — was reformulated in the 20th by Oscar Zariski, André Weil, Jean-Pierre Serre, and decisively by Alexander Grothendieck's schemes (1960s).

The Weil conjectures (1949), proved by Pierre Deligne in 1973–74, connected the number of points on a variety over a finite field to topological invariants of the corresponding complex manifold. Modern algebraic geometry is the central organising language of much of pure mathematics.

Chapter XXVThe Langlands program.

In a 17-page handwritten letter to André Weil in January 1967, the 30-year-old Robert Langlands sketched a web of conjectures connecting Galois representations, automorphic forms, and L-functions. "If you are willing to read it as pure speculation I would appreciate that; if not — I am sure you have a waste basket handy."

It is now the central organising vision of modern number theory. The Langlands correspondence predicts a precise dictionary between two seemingly different worlds: the arithmetic side (Galois groups of number fields) and the analytic side (automorphic representations of reductive groups).

Major confirmations: Andrew Wiles's 1995 proof of the Taniyama–Shimura modularity conjecture (and hence Fermat's last theorem) is a Langlands result for elliptic curves over Q. The geometric Langlands program — Kapustin and Witten 2007 — relates Langlands to gauge theory. Laurent Lafforgue and Vincent Lafforgue won Fields Medals in 2002 and Breakthrough Prizes for further pieces.

Chapter XXVITwenty essentials.

• c.820Al-Jabr wa'l-Muqabalaal-Khwarizmi
• 1545Ars MagnaCardano
• 1591In artem analyticem isagogeViète
• 1637La GéométrieDescartes
• 1799Disquisitiones ArithmeticaeGauss
• 1854Laws of ThoughtBoole
• 1870Traité des substitutionsJordan
• 1930Moderne Algebravan der Waerden
• 1942Survey of Modern AlgebraBirkhoff & Mac Lane
• 1965Topics in AlgebraHerstein
• 1971Categories for the Working MathematicianMac Lane
• 1974AlgebraLang
• 1991Abstract AlgebraDummit & Foote
• 1995Algebraic GeometryHartshorne
• 1995Linear Algebra Done RightAxler
• 1997Galois TheoryStewart
• 2003Lie Groups, Lie Algebras, and RepresentationsHall
• 2009An Introduction to the Langlands ProgramBernstein & Gelbart
• 2013Love and MathFrenkel
• 2016Linear Algebra and Learning from DataStrang

Chapter XXVIIWatch & read.

↑ 3Blue1Brown · Cramer's rule, geometrically · Essence of Linear Algebra

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Chapter XXVIIIWhere to begin.

If you remember high-school algebra and want one book to anchor everything else: Axler's Linear Algebra Done Right. It builds the subject without determinants, in a way that prepares you for everything later.

For abstract algebra: Dummit & Foote if you want comprehensive, Herstein if you want elegant, Lang if you want terse.

For history and motivation: Edward Frenkel's Love and Math is the rare popular book by a working researcher that conveys what modern algebra is actually about. John Stillwell's Mathematics and Its History is the best single-volume tour of how the subject grew.

The reward of going deep is not the formulas. It is the recognition that nearly every mathematical structure — including the ones that turn up in physics, computer science, and statistics — is a special case of something in algebra.