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))×	Relativistic addition and group theory	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	Dihedral groups I	Dihedral groups II
Isometry group of the plane (via complex numbers)	Isometry group of Rn	SL2(R)	SL2(Z)
Escher's Print Gallery and quotient groups	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
Ring Theory
Division theorem in Z and R[T]	Standard definitions	Remarks about Euclidean domains	Symmetric polynomials
Applications of unique factorization	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	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)
Dual modules	Universal Identities I	Universal Identities II	Splitting of short exact sequences for modules
Tensor products I	Tensor products II	Complexification	Exterior Powers
Base Extension and Exterior Powers	Related aligned bases	Stably free modules	Bilinear forms
Fields and Galois theory
Roots and irreducible polynomials	Roots on the unit circle	Finite fields	Trace and norm
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
Divisibility and greatest common divisor	Pythagorean triples	Modular arithmetic	Modular arithmetic (short version)
Universal divisibility test	Fermat's compositeness test	Euler's theorem	Analogies between Z and F[T]
Gaussian integers	Fermat's method of descent	An example of descent by Euler	Congruent number problem
Arithmetic progressions of three squares	Arithmetic progressions of four squares	Quadratic reciprocity in odd characteristic	Quadratic reciprocity in characteristic 2
Euclidean proofs of Dirichlet's theorem	Sums of two squares and lattices	Factoring in quadratic fields	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
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)
Analysis
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