Group Theory

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))^{×}

Homomorphisms

No subgroup of index 2 in A_{4}

Groups of order 12

Groups of order p^{3}

Generalized Quaternions

Generating sets

Cosets and Lagrange's Theorem

Conjugation

A 2parameter nonabelian group

Dihedral groups I

Dihedral groups II

Isometry group of
the plane
(via complex numbers)

Isometry group of
R^{n}

SL_{2}(R)

SL_{2}(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 A_{n}

Simplicity of PSL_{n}(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

SchurZassenhaus 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]

Counting Roots of Polynomials

Standard definitions

Remarks about
Euclidean domains

Symmetric polynomials

Applications of unique factorization

Quaternion Algebras

Primitive vectors and SL_{n}

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 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)

Noetherian modules

Dual modules

Modules over a PID

Simultaneously 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 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

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

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

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 reciprocity
in
odd characteristic

Quadratic reciprocity
in
characteristic 2

padic 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

The ring of integers in a radical extension

A nonfree relative integral extension

Examples of class group calculations

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 different ideal

The conductor ideal

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

Hensel's lemma

Kummer's lemma

Fermat's last theorem for regular primes

Selmer's Example

Truncated ArtinHasse series and roots of unity

The character group of Q

Local compactness of Pontryagin dual group

Compact Subgroups of GL_{n}(Q_{p})

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

Probability distributions and maximum entropy

Metric spaces

The contraction mapping theorem

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



