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/(pa) × Z/(pb) 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 An Simplicity of PSLn(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 SLn 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 Sn and An 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 Fp[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 GLn(Q) The character group of Q Field automorphisms of R and Qp
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 GLn(Qp) Compact subgroups of GLn(Qp) 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
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) Lp spaces for 0 < p < 1
Fundamental Theorem of Algebra
Proof by multivariable calculus Proof by linear algebra Proof by proper maps
Spaces that are connected but not path connected Finite-dimensional topological vector spaces