Writing Proofs

Advice on mathematical writing

Examples of proofs by induction

Proofs of integrality of binomial coefficients

Welldefined 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 A_{4} has index 2

Groups of order 4 and 6

Groups of order 12

Groups of order p^{2}

Groups of order p^{3}

Groups of order 16

Generalized quaternions

Generating sets

Conjugation in a group

A 2parameter nonabelian group

Dihedral groups I

Dihedral groups II

Isometries of
the plane
and complex numbers

Isometries of
the plane
and linear algebra

Isometries of
R^{n}

SL_{2}(R)

SL_{2}(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

SchurZassenhaus 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 x^{n}  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

Orthogonal group of x^{2} + y^{2}  z^{2}

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

Infinitedimensional dual spaces

Bilinear forms

Universal identities I

Universal identities II

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

ArtinSchreier 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

Square patterns and infinitely many primes

The "topological" proof of the infinitude of primes

SolovayStrassen test

MillerRabin 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

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 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 nonfree relative integral extension

Ideal classes and Kronecker bound

Class group calculations by Minkowski bound

Ideal classes and relative integers

Ideal classes and SL_{2}

Ideal classes and matrix conjugation over Z

Dirichlet's unit theorem

Chevalley's unit theorem

Existence of Frobenius elements (d'après Frobenius)

Primes of degree 1 and congruence conditions

Euclidean proofs of Dirichlet's theorem

Irreducibility of truncated exponentials

The Galois group of x^{n}  x  1 over Q

The different ideal

The conductor ideal of an order

Lfunctions 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 padic expansion of rational numbers

Binomial coefficients and padic limits

padic harmonic sums

Hensel's lemma

A multivariable Hensel's lemma

Equivalence of absolute values

Equivalence of norms

The localglobal principle

Primepower units and finite subgroups of GL_{n}(Q)

The character group of Q

Field automorphisms of R and Q_{p}

Infinite series in padic fields

Mahler expansions

An application of Strassmann's theorem

Integral solutions of x^{3}  2y^{3} = 1.

Truncated ArtinHasse series and roots of unity

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

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

The remainder in Taylor series

Abel's theorem

Accelerating convergence of series

Arc length, integration by parts, and π

Irrationality of π and e

Probability distributions and maximum entropy

Metric spaces

The contraction mapping theorem

The contraction mapping theorem, II

The space c_{0}(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

Finitedimensional topological vector spaces


