Writing Proofs
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Advice on mathematical writing
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Examples of proofs by induction
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Proofs of integrality of binomial coefficients
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Well-defined functions
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Group Theory
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Why groups?
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Sign of permutations
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The Fifteen puzzle (and Rubik's cube)
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Order of elements
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Subgroups of cyclic groups
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Subgroups of Z/(pa) × Z/(pb)
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Cyclicity of (Z/(p))×
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Cosets and Lagrange's theorem
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Quotient groups
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Homomorphisms
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Isomorphisms
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No subgroup of A4 has index 2
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Groups of order 4 and 6
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Groups of order 12
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Groups of order p2
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Groups of order p3
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Groups of order 16
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Generalized quaternions
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Generating sets
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Conjugation in a group
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A 2-parameter nonabelian group
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Dihedral groups I
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Dihedral groups II
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Isometries of
the plane
and complex numbers
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Isometries of
the plane
and linear algebra
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Isometries of
Rn
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SL2(R)
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SL2(Z)
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Proof of Cauchy's theorem
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Consequences of Cauchy's theorem
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Decomposition of finite abelian groups
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Group actions
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Transitive group actions
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The Sylow theorems (proof)
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Consequences of Sylow theorems
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More on the Sylow theorems
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When are all groups of order n cyclic?
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Simplicity of An
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Simplicity of PSLn(F)
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Characters of finite abelian groups
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Characters of finite abelian groups (short version)
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Semidirect Products
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Subgroup series I
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Subgroup series II
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Splitting of short exact sequences for groups
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Schur-Zassenhaus theorem
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Relativistic addition and group theory
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Escher's Print Gallery
and quotient groups
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Maschke's theorem over general fields
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Representations of affine and Heisenberg group over finite fields
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The degree may not divide the size of the group
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Why word problems are hard
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Ring Theory
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Division theorem in Z and R[T]
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Counting roots of polynomials
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Standard definitions for rings
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Notes on ideals
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Irreducibility tests in Q[T]
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An irreducible that factors modulo all primes
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Irreducibility of xn - x - 1
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The Gauss norm and Gauss's lemma
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Remarks about
Euclidean domains
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Noetherian rings
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Symmetric polynomials
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Applications of unique factorization
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Nilpotents, units, and zero divisors for polynomials
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Maximal ideals in polynomial rings
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Primitive vectors and SLn
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Zorn's lemma
(in group theory, ring theory, and linear algebra)
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Algebras
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Quaternion algebras
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Linear/Multilinear algebra
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Dimension
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Minimal polynomial
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Simultaneous commutativity of operators
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Potentially diagonalizable operators
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Semisimple operators
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Differential equations and linear algebra
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Linear recursions over all fields
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The norm of a matrix
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Pythagorean descent
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Pfister's theorem on sums of squares
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Hurwitz's theorem on sums of squares
(by
linear algebra)
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Hurwitz's theorem on sums of squares
(by
representation theory)
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Sums of squares in Q and F(T)
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Introduction to modules
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Modules over a PID
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Simultaneously aligned bases
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Stably free modules
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Noetherian modules
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Dual modules
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Infinite-dimensional dual spaces
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Bilinear forms
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Universal identities I
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Universal identities II
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Universal mapping properties
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Splitting of short exact sequences for modules
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Complexification
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Tensor products I
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Tensor products II
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Exterior powers
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Base extension and exterior powers
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Fields and Galois theory
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Roots and irreducible polynomials
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Roots on a circle
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Simple radical extensions
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Finite fields
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Trace and norm, I
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Trace and norm, II
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Separable extensions
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Perfect fields
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Constructing algebraic closures, I
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Constructing algebraic closures, II
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Zorn's lemma
(with fields)
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Splitting fields
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Separable extensions and tensor products
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Splitting fields and tensor products
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Galois correspondence
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Examples of Galois correspondence
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Applications of Galois theory
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Galois groups as permutation groups
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Galois correspondence theorems
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Galois groups of cubics and quartics (not char. 2)
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Galois groups of cubics and quartics (all characteristics)
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Cyclotomic extensions
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Recognizing Galois groups Sn and An
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Linear independence of characters
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The Artin-Schreier theorem
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Galois descent
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Elementary number theory
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The division theorem in Z and F[T]
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Divisibility and greatest common divisor
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Divisibility without Bezout's identity
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Modular arithmetic
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Modular arithmetic (short version)
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Unique factorization in Z and F[T]
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Analogies between Z and F[T]
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Universal divisibility test
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Pythagorean triples
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Fermat's little theorem
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Fermat's test
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Euler's theorem
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Orders in modular arithmetic
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Number theory and cryptography
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Chinese remainder theorem
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Carmichael numbers and Korselt's criterion
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When is -1 a square modulo primes?
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The infinitude of the primes
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Patterns in primes
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Wieferich primes
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Square patterns and infinitely many primes
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The "topological" proof of the infinitude of primes
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Solovay-Strassen test
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Miller-Rabin test
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Irreducibility tests in Fp[T]
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Pell's equation, I
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Pell's equation, II
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Negation and inversion of continued fractions
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Gaussian integers
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Factoring in quadratic fields
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Sums of two squares and lattices
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Proofs by descent
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An example of
descent by Euler
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Congruent number problem
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Arithmetic progressions of three squares
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Arithmetic progressions of four squares
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Quadratic residue patterns modulo a prime
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Quadratic reciprocity in
odd characteristic
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Quadratic reciprocity
in
characteristic 2
|
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Algebraic number theory
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Examples of
Mordell's equation
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Factoring in quadratic fields
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Unique factorization of ideals
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Factoring ideals after Dedekind
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Dedekind's index theorem
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Discriminants and ramified primes
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Totally ramified primes and Eisenstein polynomials
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Rings of integers without a power basis
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The ring of integers in a radical extension
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A non-free relative integral extension
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Ideal classes and Kronecker bound
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Class group calculations by Minkowski bound
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Ideal classes and relative integers
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Ideal classes and SL2
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Ideal classes and matrix conjugation over Z
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Dirichlet's unit theorem
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Chevalley's unit theorem
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Existence of Frobenius elements (d'après Frobenius)
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Galois groups over Q and factorizations mod p
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Dedekind domains and Galois residue field extensions
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Primes of degree 1 and congruence conditions
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Euclidean proofs of Dirichlet's theorem
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Irreducibility of truncated exponentials
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The Galois group of xn - x - 1 over Q
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The different ideal
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The conductor ideal of an order
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Gauss and Jacobi sums on finite fields and Z/mZ
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L-functions for Gauss and Jacobi sums
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Invariants of the splitting field of a cubic, I
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Invariants of the splitting field of a cubic, II
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Invariants of the splitting field of a cubic, III
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Invariants of the splitting field of a cubic, IV
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Invariants of the splitting field of a cubic, V
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Ostrowski's theorem for
Q
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Ostrowski's theorem for
Q(i)
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Ostrowski's theorem for
F(T)
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Ostrowski's theorem for
number fields
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The p-adic expansion of rational numbers
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Binomial coefficients and p-adic limits
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p-adic harmonic sums
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Hensel's lemma
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A multivariable Hensel's lemma
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Equivalence of absolute values
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Equivalence of norms
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The local-global principle
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Prime-power units and finite subgroups of GLn(Q)
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The character group of Q
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Field automorphisms of R and Qp
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Infinite series in p-adic fields
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Mahler expansions
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An application of Strassmann's theorem
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Integral solutions of x3 - 2y3 = 1.
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Truncated Artin-Hasse series and roots of unity
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Maximal compact subgroups of GLn(Qp)
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Compact subgroups of GLn(Qp)
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A separable extension with inseparable residue field
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Local compactness of Pontryagin dual group
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Selmer's example
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Kummer's lemma
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Fermat's last theorem for regular primes
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Carlitz extensions
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History of class field theory
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Analysis
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The Mean Value Theorem and integral powers
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Orders of growth
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Estimating growth of divergent series
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Asymptotic growth
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Stirling's formula
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Infinite series
|
The Gaussian integral
|
Estimating definite integrals of e-t2 in two ways
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The logarithm and arctangent
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The remainder in Taylor series
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Abel's theorem
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Accelerating convergence of series
|
Arc length, integration by parts, and π
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Irrationality of π and e
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Transcendence of e
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Differentiation under the integral sign
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Probability distributions and maximum entropy
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Metric spaces
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The contraction mapping theorem
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The contraction mapping theorem, II
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The space c0(K)
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Lp spaces for 0 < p < 1
|
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Fundamental Theorem of Algebra
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Proof by multivariable calculus
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Proof by linear algebra
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Proof by proper maps
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Topology
|
Spaces that are connected but not path connected
|
Finite-dimensional topological vector spaces
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