Expository papers

These were written up for various reasons: course handouts, notes to accompany a talk for a (mathematically) general audience, or for some other purpose that I have since forgotten. If you find typographical or other errors in these files, or have comments, please let me know. Files that are revised will be reposted without any indication that they have been changed (sorry).

Writing Proofs

Advice on mathematical writing Examples of proofs by induction Proofs of integrality of binomial coefficients Well-defined functions

Group Theory

Why groups? Sign of permutations The Fifteen puzzle (and Rubik's cube) Order of elements

Subgroups of cyclic groups Subgroups of Z/(p^a) × Z/(p^b) Cyclicity of (Z/(p))^× Cosets and Lagrange's theorem

Quotient groups Homomorphisms Isomorphisms No subgroup of A₄ has index 2

Groups of order 4 and 6 Groups of order 12 Groups of order p² Groups of order p³

Groups of order 16 Generalized quaternions Generating sets Conjugation in a group

A 2-parameter nonabelian group Dihedral groups I Dihedral groups II Isometries of the plane and complex numbers

Isometries of the plane and linear algebra Isometries of Rⁿ SL₂(R) SL₂(Z)

Proof of Cauchy's theorem Consequences of Cauchy's theorem Decomposition of finite abelian groups Group actions

Transitive group actions The Sylow theorems (proof) Consequences of Sylow theorems More on the Sylow theorems

When are all groups of order n cyclic? Simplicity of A_n Simplicity of PSL_n(F) Characters of finite abelian groups

Characters of finite abelian groups (short version) Semidirect Products Subgroup series I Subgroup series II

Splitting of short exact sequences for groups Schur-Zassenhaus theorem Relativistic addition and group theory Escher's Print Gallery and quotient groups

Maschke's theorem over general fields Representations of affine and Heisenberg group over finite fields The degree may not divide the size of the group Why word problems are hard

Ring Theory

Division theorem in Z and R[T] Counting roots of polynomials Standard definitions for rings Notes on ideals

Irreducibility tests in Q[T] An irreducible that factors modulo all primes Irreducibility of xⁿ - x - 1 The Gauss norm and Gauss's lemma

Remarks about Euclidean domains Noetherian rings Symmetric polynomials Applications of unique factorization

Nilpotents, units, and zero divisors for polynomials Maximal ideals in polynomial rings Primitive vectors and SL_n Zorn's lemma (in group theory, ring theory, and linear algebra)

Algebras Quaternion algebras

Linear/Multilinear algebra

Dimension Minimal polynomial Simultaneous commutativity of operators Potentially diagonalizable operators

Semisimple operators Differential equations and linear algebra Linear recursions over all fields The norm of a matrix

Pythagorean descent Pfister's theorem on sums of squares Hurwitz's theorem on sums of squares (by linear algebra) Hurwitz's theorem on sums of squares (by representation theory)

Sums of squares in Q and F(T) Introduction to modules Modules over a PID Simultaneously aligned bases

Stably free modules Noetherian modules Dual modules Infinite-dimensional dual spaces

Bilinear forms Universal identities I Universal identities II Universal mapping properties

Splitting of short exact sequences for modules Complexification Tensor products I Tensor products II

Exterior powers Base extension and exterior powers

Fields and Galois theory

Roots and irreducible polynomials Roots on a circle Simple radical extensions Finite fields

Trace and norm, I Trace and norm, II Separable extensions Perfect fields

Constructing algebraic closures, I Constructing algebraic closures, II Zorn's lemma (with fields) Splitting fields

Separable extensions and tensor products Splitting fields and tensor products Galois correspondence Examples of Galois correspondence

Applications of Galois theory Galois groups as permutation groups Galois correspondence theorems Galois groups of cubics and quartics (not char. 2)

Galois groups of cubics and quartics (all characteristics) Cyclotomic extensions Recognizing Galois groups S_n and A_n Linear independence of characters

The Artin-Schreier theorem Galois descent

Elementary number theory

The division theorem in Z and F[T] Divisibility and greatest common divisor Divisibility without Bezout's identity Modular arithmetic

Modular arithmetic (short version) Unique factorization in Z and F[T] Analogies between Z and F[T] Universal divisibility test

Pythagorean triples Fermat's little theorem Fermat's test Euler's theorem

Orders in modular arithmetic Number theory and cryptography Chinese remainder theorem Carmichael numbers and Korselt's criterion

When is -1 a square modulo primes? The infinitude of the primes Patterns in primes Wieferich primes

Square patterns and infinitely many primes The "topological" proof of the infinitude of primes Solovay-Strassen test Miller-Rabin test

Irreducibility tests in F_p[T] Pell's equation, I Pell's equation, II Negation and inversion of continued fractions

Gaussian integers Factoring in quadratic fields Sums of two squares and lattices Proofs by descent

An example of descent by Euler Congruent number problem Arithmetic progressions of three squares Arithmetic progressions of four squares

Quadratic residue patterns modulo a prime Quadratic reciprocity in odd characteristic Quadratic reciprocity in characteristic 2

Algebraic number theory

Examples of Mordell's equation Factoring in quadratic fields Unique factorization of ideals Factoring ideals after Dedekind

Dedekind's index theorem Discriminants and ramified primes Totally ramified primes and Eisenstein polynomials Rings of integers without a power basis

The ring of integers in a radical extension A non-free relative integral extension Ideal classes and Kronecker bound Class group calculations by Minkowski bound

Ideal classes and relative integers
Ideal classes and SL₂
Ideal classes and matrix conjugation over Z Dirichlet's unit theorem

Chevalley's unit theorem Existence of Frobenius elements (d'après Frobenius) Galois groups over Q and factorizations mod p Dedekind domains and Galois residue field extensions

Primes of degree 1 and congruence conditions Euclidean proofs of Dirichlet's theorem Irreducibility of truncated exponentials The Galois group of xⁿ - x - 1 over Q

The different ideal The conductor ideal of an order Gauss and Jacobi sums on finite fields and Z/mZ L-functions for Gauss and Jacobi sums

Invariants of the splitting field of a cubic, I Invariants of the splitting field of a cubic, II Invariants of the splitting field of a cubic, III Invariants of the splitting field of a cubic, IV

Invariants of the splitting field of a cubic, V Ostrowski's theorem for Q Ostrowski's theorem for Q(i) Ostrowski's theorem for F(T)

Ostrowski's theorem for number fields The p-adic expansion of rational numbers Binomial coefficients and p-adic limits p-adic harmonic sums

Hensel's lemma A multivariable Hensel's lemma Equivalence of absolute values Equivalence of norms

The local-global principle Prime-power units and finite subgroups of GL_n(Q) The character group of Q Field automorphisms of R and Q_p

Infinite series in p-adic fields Mahler expansions An application of Strassmann's theorem Integral solutions of x³ - 2y³ = 1.

Truncated Artin-Hasse series and roots of unity Maximal compact subgroups of GL_n(Q_p) Compact subgroups of GL_n(Q_p) A separable extension with inseparable residue field

Local compactness of Pontryagin dual group Selmer's example Kummer's lemma Fermat's last theorem for regular primes

Carlitz extensions History of class field theory

Analysis

The Mean Value Theorem and integral powers Orders of growth Estimating growth of divergent series Asymptotic growth

Stirling's formula Infinite series The Gaussian integral Estimating definite integrals of e^-t² in two ways

The logarithm and arctangent The remainder in Taylor series Abel's theorem Accelerating convergence of series

Arc length, integration by parts, and π Irrationality of π and e Transcendence of e Differentiation under the integral sign

Probability distributions and maximum entropy Metric spaces The contraction mapping theorem The contraction mapping theorem, II

The space c₀(K) L^p spaces for 0 < p < 1

Fundamental Theorem of Algebra

Proof by multivariable calculus Proof by linear algebra Proof by proper maps

Topology

Spaces that are connected but not path connected Finite-dimensional topological vector spaces

Writing Proofs
Advice on mathematical writing	Examples of proofs by induction	Proofs of integrality of binomial coefficients	Well-defined functions
Group Theory
Why groups?	Sign of permutations	The Fifteen puzzle (and Rubik's cube)	Order of elements
Subgroups of cyclic groups	Subgroups of Z/(p^a) × Z/(p^b)	Cyclicity of (Z/(p))^×	Cosets and Lagrange's theorem
Quotient groups	Homomorphisms	Isomorphisms	No subgroup of A₄ has index 2
Groups of order 4 and 6	Groups of order 12	Groups of order p²	Groups of order p³
Groups of order 16	Generalized quaternions	Generating sets	Conjugation in a group
A 2-parameter nonabelian group	Dihedral groups I	Dihedral groups II	Isometries of the plane and complex numbers
Isometries of the plane and linear algebra	Isometries of Rⁿ	SL₂(R)	SL₂(Z)
Proof of Cauchy's theorem	Consequences of Cauchy's theorem	Decomposition of finite abelian groups	Group actions
Transitive group actions	The Sylow theorems (proof)	Consequences of Sylow theorems	More on the Sylow theorems
When are all groups of order n cyclic?	Simplicity of A_n	Simplicity of PSL_n(F)	Characters of finite abelian groups
Characters of finite abelian groups (short version)	Semidirect Products	Subgroup series I	Subgroup series II
Splitting of short exact sequences for groups	Schur-Zassenhaus theorem	Relativistic addition and group theory	Escher's Print Gallery and quotient groups
Maschke's theorem over general fields	Representations of affine and Heisenberg group over finite fields	The degree may not divide the size of the group	Why word problems are hard
Ring Theory
Division theorem in Z and R[T]	Counting roots of polynomials	Standard definitions for rings	Notes on ideals
Irreducibility tests in Q[T]	An irreducible that factors modulo all primes	Irreducibility of xⁿ - x - 1	The Gauss norm and Gauss's lemma
Remarks about Euclidean domains	Noetherian rings	Symmetric polynomials	Applications of unique factorization
Nilpotents, units, and zero divisors for polynomials	Maximal ideals in polynomial rings	Primitive vectors and SL_n	Zorn's lemma (in group theory, ring theory, and linear algebra)
Algebras	Quaternion algebras
Linear/Multilinear algebra
Dimension	Minimal polynomial	Simultaneous commutativity of operators	Potentially diagonalizable operators
Semisimple operators	Differential equations and linear algebra	Linear recursions over all fields	The norm of a matrix
Pythagorean descent	Pfister's theorem on sums of squares	Hurwitz's theorem on sums of squares (by linear algebra)	Hurwitz's theorem on sums of squares (by representation theory)
Sums of squares in Q and F(T)	Introduction to modules	Modules over a PID	Simultaneously aligned bases
Stably free modules	Noetherian modules	Dual modules	Infinite-dimensional dual spaces
Bilinear forms	Universal identities I	Universal identities II	Universal mapping properties
Splitting of short exact sequences for modules	Complexification	Tensor products I	Tensor products II
Exterior powers	Base extension and exterior powers
Fields and Galois theory
Roots and irreducible polynomials	Roots on a circle	Simple radical extensions	Finite fields
Trace and norm, I	Trace and norm, II	Separable extensions	Perfect fields
Constructing algebraic closures, I	Constructing algebraic closures, II	Zorn's lemma (with fields)	Splitting fields
Separable extensions and tensor products	Splitting fields and tensor products	Galois correspondence	Examples of Galois correspondence
Applications of Galois theory	Galois groups as permutation groups	Galois correspondence theorems	Galois groups of cubics and quartics (not char. 2)
Galois groups of cubics and quartics (all characteristics)	Cyclotomic extensions	Recognizing Galois groups S_n and A_n	Linear independence of characters
The Artin-Schreier theorem	Galois descent
Elementary number theory
The division theorem in Z and F[T]	Divisibility and greatest common divisor	Divisibility without Bezout's identity	Modular arithmetic
Modular arithmetic (short version)	Unique factorization in Z and F[T]	Analogies between Z and F[T]	Universal divisibility test
Pythagorean triples	Fermat's little theorem	Fermat's test	Euler's theorem
Orders in modular arithmetic	Number theory and cryptography	Chinese remainder theorem	Carmichael numbers and Korselt's criterion
When is -1 a square modulo primes?	The infinitude of the primes	Patterns in primes	Wieferich primes
Square patterns and infinitely many primes	The "topological" proof of the infinitude of primes	Solovay-Strassen test	Miller-Rabin test
Irreducibility tests in F_p[T]	Pell's equation, I	Pell's equation, II	Negation and inversion of continued fractions
Gaussian integers	Factoring in quadratic fields	Sums of two squares and lattices	Proofs by descent
An example of descent by Euler	Congruent number problem	Arithmetic progressions of three squares	Arithmetic progressions of four squares
Quadratic residue patterns modulo a prime	Quadratic reciprocity in odd characteristic	Quadratic reciprocity in characteristic 2
Algebraic number theory
Examples of Mordell's equation	Factoring in quadratic fields	Unique factorization of ideals	Factoring ideals after Dedekind
Dedekind's index theorem	Discriminants and ramified primes	Totally ramified primes and Eisenstein polynomials	Rings of integers without a power basis
The ring of integers in a radical extension	A non-free relative integral extension	Ideal classes and Kronecker bound	Class group calculations by Minkowski bound
Ideal classes and relative integers	Ideal classes and SL₂	Ideal classes and matrix conjugation over Z	Dirichlet's unit theorem
Chevalley's unit theorem	Existence of Frobenius elements (d'après Frobenius)	Galois groups over Q and factorizations mod p	Dedekind domains and Galois residue field extensions
Primes of degree 1 and congruence conditions	Euclidean proofs of Dirichlet's theorem	Irreducibility of truncated exponentials	The Galois group of xⁿ - x - 1 over Q
The different ideal	The conductor ideal of an order	Gauss and Jacobi sums on finite fields and Z/mZ	L-functions for Gauss and Jacobi sums
Invariants of the splitting field of a cubic, I	Invariants of the splitting field of a cubic, II	Invariants of the splitting field of a cubic, III	Invariants of the splitting field of a cubic, IV
Invariants of the splitting field of a cubic, V	Ostrowski's theorem for Q	Ostrowski's theorem for Q(i)	Ostrowski's theorem for F(T)
Ostrowski's theorem for number fields	The p-adic expansion of rational numbers	Binomial coefficients and p-adic limits	p-adic harmonic sums
Hensel's lemma	A multivariable Hensel's lemma	Equivalence of absolute values	Equivalence of norms
The local-global principle	Prime-power units and finite subgroups of GL_n(Q)	The character group of Q	Field automorphisms of R and Q_p
Infinite series in p-adic fields	Mahler expansions	An application of Strassmann's theorem	Integral solutions of x³ - 2y³ = 1.
Truncated Artin-Hasse series and roots of unity	Maximal compact subgroups of GL_n(Q_p)	Compact subgroups of GL_n(Q_p)	A separable extension with inseparable residue field
Local compactness of Pontryagin dual group	Selmer's example	Kummer's lemma	Fermat's last theorem for regular primes
Carlitz extensions	History of class field theory
Analysis
The Mean Value Theorem and integral powers	Orders of growth	Estimating growth of divergent series	Asymptotic growth
Stirling's formula	Infinite series	The Gaussian integral	Estimating definite integrals of e^-t² in two ways
The logarithm and arctangent	The remainder in Taylor series	Abel's theorem	Accelerating convergence of series
Arc length, integration by parts, and π	Irrationality of π and e	Transcendence of e	Differentiation under the integral sign
Probability distributions and maximum entropy	Metric spaces	The contraction mapping theorem	The contraction mapping theorem, II
The space c₀(K)	L^p spaces for 0 < p < 1
Fundamental Theorem of Algebra
Proof by multivariable calculus	Proof by linear algebra	Proof by proper maps
Topology
Spaces that are connected but not path connected	Finite-dimensional topological vector spaces