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
Cyclicity of (Z/(p))×	Relativistic addition and group theory	Cosets	Conjugation
Dihedral groups I	Dihedral groups II	Isometry group of the plane (via complex numbers)	Isometry group of Rn
Proof of Cauchy's theorem	Applications of Cauchy's theorem	Group actions	Transitive group actions
Generating sets	No subgroup of index 2 in A4	Groups of order 12	Groups of order p3
Generalized Quaternions	Splitting of short exact sequences for groups	SL2(R)	SL2(Z)
Proof of Sylow theorems	Applications of Sylow theorems	More on Sylow theorems	Subgroup series
Characters of finite abelian groups	Simplicity of An	Schur-Zassenhaus theorem	The degree may not divide the size of the group
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	The fundamental theorem of algebra via linear algebra	Differential equations and linear algebra	The norm of a matrix
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	A nonfree stably free module	Bilinear forms
Orthogonal group of x2 + y2 - z2
Fields and Galois theory
Roots and irreducible polynomials	Polynomials over a finite field	Trace and norm	Separable extensions
Perfect fields	Algebraic closures	Zorn's lemma (with fields)	Separable extensions and tensor products
Splitting fields and tensor products	Galois correspondence	Applications of Galois theory	Cyclotomic extensions
Artin-Schreier theorem	Linear independence of characters	Galois descent
Elementary number theory
Divisibilty and greatest common divisor	Pythagorean triples	Modular arithmetic	Universal divisibility test
Fermat's compositeness test	Euler's theorem	Heuristics for prime statistics	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 rings
Algebraic number theory
Examples of Mordell's equation	Factoring in quadratic rings	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	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
History of class field theory	The character group of Q	Local compactness of Pontryagin dual group
Analysis
The fundamental theorem of algebra via multivariable calculus	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