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).

Group Theory
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))× Homomorphisms No subgroup of index 2 in A4
Groups of order 12 Groups of order p3 Generalized Quaternions Generating sets
Cosets and Lagrange's Theorem Conjugation A 2-parameter nonabelian group Dihedral groups I
Dihedral groups II Isometry group of the plane (via complex numbers) Isometry group of Rn SL2(R)
SL2(Z) Proof of Cauchy's theorem Applications of Cauchy's theorem Group actions
Transitive group actions Proof of Sylow theorems Applications of Sylow theorems More on Sylow theorems
Simplicity of An Simplicity of PSLn(F) Characters of finite abelian groups Characters of finite abelian groups (short version)
Subgroup series I Subgroup series II Splitting of short exact sequences for groups Schur-Zassenhaus theorem
The degree may not divide the size of the group Representations of affine and Heisenberg group over finite fields Relativistic addition and group theory Escher's Print Gallery and quotient groups
Ring Theory
Division theorem in Z and R[T] Standard definitions Remarks about Euclidean domains Symmetric polynomials
Applications of unique factorization Quaternion Algebras Primitive vectors and SLn Zorn's lemma (in group theory, ring theory, and linear algebra)
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
Orthogonal group of x2 + y2 - z2 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) Noetherian modules Dual modules Modules over a PID
Related aligned bases Stably free modules Universal Identities I Universal Identities II
Splitting of short exact sequences for modules Complexification Bilinear forms Tensor products I
Tensor products II Exterior Powers Base Extension and Exterior Powers
Fields and Galois theory
Roots and irreducible polynomials Roots on the unit circle 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) 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 Artin-Schreier theorem Galois descent
Elementary number theory
The Division Theorem in Z and F[T] Divisibility and greatest common divisor 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 Pell's Equation, I Pell's Equation, II
Gaussian integers Factoring in quadratic fields Solovay-Strassen Test Miller-Rabin Test
Irreducibility Tests in Fp[T] Sums of two squares and lattices Fermat's method of descent An example of descent by Euler
Congruent number problem Arithmetic progressions of three squares Arithmetic progressions of four squares Euclidean proofs of Dirichlet's theorem
Quadratic reciprocity in odd characteristic Quadratic reciprocity in characteristic 2 p-adic Harmonic Sums
Algebraic number theory
Examples of Mordell's equation Factoring in quadratic fields Unique factorization of ideals Factoring ideals after Dedekind
Discriminants and ramified primes Totally ramified primes and Eisenstein polynomials Rings of integers without a power basis A non-free relative integral extension
Ideal classes and SL2
Ideal classes and matrix conjugation over Z Examples of class group calculations The different ideal
Dirichlet's unit theorem Chevalley's unit theorem Existence of Frobenius elements (d'après Frobenius) Primes of degree 1 and congruence conditions
The conductor ideal Carlitz extensions Hensel's lemma Ostrowski's theorem for Q(i)
Ostrowski for F(T) Ostrowski for number fields Irreducibility of Truncated Exponentials 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
Kummer's lemma Fermat's last theorem for regular primes Selmer's Example History of class field theory
The character group of Q Local compactness of Pontryagin dual group Compact Subgroups of GLn(Qp)
Orders of growth Estimating growth of divergent series Asymptotic growth Stirling's formula
The Gaussian integral Differentiation under the integral sign Infinite series The logarithm and arctangent
Abel's theorem The contraction mapping theorem Probability distributions and maximum entropy 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