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# Nav/Statement

The following table lists all of the statements in the wiki.

Page Link Title Statement Type Clusters
State/A OR B is logically equivalent to NOT A IMPLIES B A OR B is logically equivalent to NOT A IMPLIES B Fact Clust/Logic and foundations
State/Additivity of polynomial degrees Additivity of polynomial degrees Proposition Clust/The algebra of polynomials
State/Fields are integral domains All fields are integral domains Proposition Clust/Basic ring theory
State/Arithmetic of residues is well-defined Arithmetic of residues is well-defined Proposition Clust/Modular arithmetic
State/Arithmetic progression rule for binary quadratic forms Arithmetic progression rule for binary quadratic forms Proposition Clust/Binary quadratic forms
State/Axiom of choice Axiom of Choice Axiom Clust/ZFC axioms of set theory
State/Axiom of extensionality Axiom of Extensionality Axiom Clust/ZFC axioms of set theory
State/Axiom of infinity Axiom of Infinity Axiom Clust/ZFC axioms of set theory
State/Axiom of pairing Axiom of Pairing Axiom Clust/ZFC axioms of set theory
State/Axiom of power sets Axiom of Power Sets. Axiom Clust/ZFC axioms of set theory
State/Axiom of regularity Axiom of Regularity Axiom Clust/ZFC axioms of set theory
State/Axiom of replacement Axiom of Replacement Axiom Clust/ZFC axioms of set theory
State/Axiom of selection Axiom of Selection Axiom Clust/ZFC axioms of set theory
State/Axiom of unions Axiom of Unions Axiom Clust/ZFC axioms of set theory
State/Axiom of the empty set Axiom of the empty set. Axiom Clust/ZFC axioms of set theory
State/Being relatively prime to a product is equivalent to being relatively prime to the factors Being relatively prime to a product is equivalent to being relatively prime to the factors Proposition Clust/Basic number theory
State/Bijections are injective and surjective functions Bijections are injective and surjective functions Proposition Clust/Functions
State/Binary quadratic forms are uniquely determined by their values at a superbasis Binary quadratic forms are uniquely determined by their values at a superbasis Proposition Clust/Binary quadratic forms
State/Bounding riverbends of a given discriminant Bounding riverbends of a given discriminant Proposition
State/Bounding wells of a given discriminant Bounding wells of a given discriminant Proposition Clust/Binary quadratic forms
State/Cancellation in groups Cancellation in groups Proposition Clust/Basic group theory
State/Canonical decompositions can be used to find GCD and LCM Canonical decompositions can be used to find GCD and LCM. Proposition Clust/Basic number theory
State/Canonical decompositions of binomial coefficients Canonical decompositions of binomial coefficients Proposition Clust/Basic number theory
State/Canonical decompositions of factorials Canonical decompositions of factorials Proposition Clust/Basic number theory
State/Centers of p-groups are nontrivial Centers of p-groups are nontrivial Proposition Clust/Structure theory of finite groups
State/Chebyshev estimates for the prime number function Chebyshev estimates for the prime number function Theorem Clust/Analytic number theory
State/Chinese remainder theorem Chinese remainder theorem Theorem Clust/Modular arithmetic
State/Classification of subgroups of the integers Classification of subgroups of \$\ZZ\$ Proposition Clust/Basic group theory
State/Climbing in topographs Climbing in topographs Proposition Clust/Binary quadratic forms
State/Composing injective surjective or bijective functions yields the same Composing injective surjective or bijective functions yields the same Proposition Clust/Functions
State/Computing the totient of a prime power Computing the totient of a prime power Proposition Clust/Modular arithmetic
State/Conjugacy of Sylow subgroups Conjugacy of Sylow subgroups Theorem Clust/Structure theory of finite groups
State/Counting Sylow subgroups Counting Sylow subgroups Theorem Clust/Structure theory of finite groups
State/Counting multiples of d between 1 and n Counting multiples of d between 1 and n Proposition Clust/Basic number theory
Clust/Basic counting
State/Counting subsets of a given cardinality Counting subsets of a given cardinality Proposition Clust/Basic counting
State/Cyclic groups are classified by their order. Cyclic groups are classified by their order. Proposition Clust/Basic group theory
State/Divisibility corresponds to containment of principal ideals Divisibility corresponds to containment of principal ideals Proposition Clust/Basic ring theory
State/Divisibility corresponds to inequalities of prime exponents Divisibility corresponds to inequalities of prime exponents Proposition Clust/Basic number theory
State/Estimate for the product of primes Estimate for the product of primes Proposition
State/Eulers criterion Euler Criterion Proposition Clust/Theory of quadratic residues
State/Every arc of river is finite Every arc of river is finite Proposition Clust/Binary quadratic forms
State/Every lax vector in a lax basis is primitive Every lax vector in a lax basis is primitive Proposition Clust/Binary quadratic forms
State/Every nonempty subset of N has a smallest element Every nonempty subset of N has a smallest element. Proposition Clust/Properties of natural numbers
State/Every primitive lax vector belongs to a lax basis Every primitive lax vector belongs to a lax basis Proposition Clust/Binary quadratic forms
State/Existence of Sylow subgroups Existence of Sylow subgroups Theorem Clust/Structure theory of finite groups
State/Existence of wells for definite forms Existence of wells for definite forms Proposition Clust/Binary quadratic forms
State/Fermats little theorem Fermat's little theorem Theorem Clust/Modular arithmetic
State/Fermat Euler theorem Fermat-Euler theorem Theorem Clust/Modular arithmetic
State/Finite sets can be counted Finite sets can be counted Proposition Clust/Naive set theory
State/First isomorphism theorem for groups First isomorphism theorem for groups Proposition Clust/Basic group theory
State/Fundamental theorem of algebra Fundamental theorem of algebra Theorem Clust/Complex analysis
Clust/Field theory
State/Greatest common divisors can be computed pairwise Greatest common divisors can be computed pairwise Proposition Clust/Basic number theory
State/Greatest common divisors can be found with the Euclidean algorithm Greatest common divisors can be found with the Euclidean algorithm Proposition Clust/Basic number theory
State/Groups of prime order are cyclic Groups of prime order are cyclic Proposition Clust/Structure theory of finite groups
State/Groups of prime squared order are abelian Groups of prime squared order are abelian Proposition Clust/Structure theory of finite groups
State/Half of nonzero residues are quadratic residues Half of nonzero residues are quadratic residues Proposition Clust/Theory of quadratic residues
State/Homomorphic images of subgroups are subgroups Homomorphic images of subgroups are subgroups Proposition
State/Induction Induction Theorem Clust/Properties of natural numbers
State/Inequality in N corresponds to existence of differences Inequality in N corresponds to existence of differences Proposition Clust/Properties of natural numbers
State/Injective functions have left inverses Injective functions have left inverses Proposition Clust/Functions
State/Integers are relatively prime iff they have no common prime factors Integers are relatively prime iff they have no common prime factors Proposition Clust/Basic number theory
State/Intersections of subspaces are subspaces Intersections of subspaces are subspaces Proposition Clust/Linear algebra
State/Inversion is an antiautomorphism Inversion is an antiautomorphism Proposition Clust/Basic group theory
State/Irreducible implies prime in a PID Irreducible elements of a PID are prime. Proposition Clust/Factorization in rings
State/Kernel criterion for injectivity of group homomorphisms Kernel criterion for injectivity of group homomorphisms Proposition Clust/Basic group theory
State/Lagranges theorem Lagrange's Theorem Theorem Clust/Basic group theory
State/Linear Diophantine equations can be solved with the Euclidean algorithm Linear Diophantine equations can be solved with the Euclidean algorithm Proposition Clust/Basic number theory
State/Multiplication increases size Multiplication increases size. Fact Clust/Arithmetic in N
State/Multiplicative inverses exist mod p Multiplicative inverses exist mod p Proposition Clust/Modular arithmetic
State/Multiplicativity of the totient Multiplicativity of the totient Proposition Clust/Modular arithmetic
State/Mutual divisibility of natural numbers implies equality Mutual divisibility of natural numbers implies equality Proposition Clust/Basic number theory
State/N is totally ordered N is totally ordered Proposition Clust/Properties of natural numbers
State/Natural numbers are prime or composite or zero or one Natural numbers are prime or composite or zero or one Proposition Clust/Basic number theory
State/Natural numbers can be factored into primes Natural numbers can be factored into primes Proposition Clust/Basic number theory
State/No set is an element of itself No set is an element of itself Proposition Clust/Basic set theory
State/Nonmultiples of a prime are relatively prime to the prime Nonmultiples of a prime are relatively prime to the prime Proposition Clust/Basic number theory
State/Nonzero naturals are successors Nonzero naturals are successors Fact Clust/Properties of natural numbers
State/Nilpotents are zero divisors Nonzero nilpotents are zero divisors Proposition Clust/Basic ring theory
State/Periodicity along the river Periodicity along the river Proposition Clust/Binary quadratic forms
State/Permutations can be decomposed into transpositions Permutations can be decomposed into transpositions Proposition Clust/Permutations
State/Primes dividing a product must divide a factor Primes dividing a product must divide a factor Proposition Clust/Basic number theory
State/Product of units is a unit Product of units is a unit Proposition Clust/Basic ring theory
State/Products of invertible residues are invertible Products of invertible residues are invertible Proposition Clust/Modular arithmetic
State/Rational roots of integers are integers Rational roots of integers are integers Theorem Clust/Basic number theory
State/Relative primality to the modulus is equivalent to invertibility of a residue Relative primality to the modulus is equivalent to invertibility of a residue Proposition Clust/Modular arithmetic
State/Root counting over a field Root counting over a field Proposition Clust/The algebra of polynomials
State/Solutions to homogeneous linear Diophantine equations can be found with the LCM Solutions to homogeneous linear Diophantine equations can be found with the LCM Proposition Clust/Basic number theory
State/Solving quadratic Diophantine equations in two variables Solving quadratic Diophantine equations in two variables Proposition Clust/Binary quadratic forms
State/Squares are nonnegative in an ordered field Squares are nonnegative in an ordered field Proposition Clust/Ordered ring theory
State/Surjective functions have right inverses Surjective functions have right inverses Proposition Clust/Functions
State/Systems of two linear Diophantine equations Systems of two linear Diophantine equations Proposition Clust/Binary quadratic forms
State/The GCD times the LCM is the product The GCD times the LCM is the product Proposition Clust/Basic number theory
State/The domain topograph has no circuits The domain topograph has no circuits Proposition Clust/Binary quadratic forms
State/The empty set is a subset of every set The empty set is a subset of every set. Proposition
State/The floor sum identity The floor sum identity Proposition Clust/Basic number theory
Clust/Basic counting
State/Intersections of ideals are ideals The intersection of ideals is an ideal Proposition Clust/Basic ring theory
State/Intersections of subgroups are subgroups The intersection of subgroups is a subgroup. Proposition Clust/Basic group theory
State/There are infinitely many prime numbers There are infinitely many prime numbers Theorem Clust/Basic number theory
State/No nonzero nilpotents in a field There are no nonzero nilpotents in a field. Proposition Clust/Basic ring theory
State/There are no zero divisors mod p There are no zero divisors mod p Proposition Clust/Modular arithmetic
State/Triangle inequalities Triangle inequalities Proposition Clust/Coordinate plane geometry
State/Truth by false premise Truth by False Premise Fact Clust/Logic and foundations
State/Two out of three principle for divisibility Two out of three principle for divisibility. Proposition Clust/Basic number theory
Clust/Basic ring theory
State/Uniqueness of identity element in a monoid Uniqueness of identity element in a monoid or group Fact Clust/Basic group theory
State/Uniqueness of prime factorization Uniqueness of prime factorization Theorem Clust/Basic number theory
State/Uniqueness of inverse elements in a monoid Uniqueness_of_inverse_elements_in_a_monoid Fact Clust/Basic group theory
State/Values in the range topograph adjacent to a given face form a quadratic sequence Values in the range topograph adjacent to a given face form a quadratic sequence Proposition Clust/Binary quadratic forms
State/Zolotarevs lemma Zolotarev's Lemma Lemma Clust/Theory of quadratic residues