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Showing 121–150 of 1709 reference entries

Mathematics & logicZFC set-theory framework

Set existence is governed by explicit axioms and schemas.

DefinitionMathematics & logicCompleteness of the real numbers

Upper-bounded nonempty real sets have least upper bounds.

Mathematical statementMathematics & logicFinite and countable sets

Countability includes finite sets in this convention.

DefinitionMathematics & logicEquivalence relations and quotient classes

An equivalence relation partitions a set into classes.

DefinitionMathematics & logicModels, completeness and incompleteness

Semantic consequence and arithmetic completeness are distinct.

Mathematical statementMathematics & logicInteger divisibility and the Euclidean algorithm

Repeated division computes a greatest common divisor.

MethodMathematics & logicUnique prime factorization of integers

Positive integers greater than one have unique prime factors.

Mathematical statementMathematics & logicInteger arithmetic modulo n

Congruence identifies integers differing by a multiple of n.

DefinitionMathematics & logicChinese Remainder Theorem

Coprime moduli specify one combined residue.

Mathematical statementMathematics & logicGroups and abelian groups

A group has associative composition, identity and inverses.

DefinitionMathematics & logicGroup homomorphisms and kernels

Homomorphisms preserve the group operation.

DefinitionMathematics & logicRings and identity conventions

A ring combines addition with distributive multiplication.

DefinitionMathematics & logicFields and division

A field allows division by every nonzero element.

DefinitionMathematics & logicFundamental Theorem of Algebra

Nonconstant complex polynomials factor into linear terms.

Mathematical statementMathematics & logicMetric spaces

A metric supplies a distance satisfying three axioms.

DefinitionMathematics & logicTopological spaces

A topology specifies which subsets are open.

DefinitionMathematics & logicCompactness and finite subcovers

Compactness reduces every open cover to a finite one.

DefinitionMathematics & logicContinuity between topological spaces

Continuity preserves openness through inverse images.

DefinitionMathematics & logicEuclidean and curved-surface geometry

Triangle-angle relations depend on the geometry.

Mathematical statementMathematics & logicAbstract vector spaces

Vector-space laws extend beyond coordinate arrows.

DefinitionMathematics & logicBases and finite dimension

A basis spans without linear redundancy.

DefinitionMathematics & logicRank–nullity theorem

Kernel and image dimensions sum to domain dimension.

Mathematical statementMathematics & logicEigenvalues and eigenvectors

An eigenvector changes only by a scalar factor.

DefinitionMathematics & logicFinite-dimensional self-adjoint spectral theorem

Self-adjoint operators have orthonormal eigenbases.

Mathematical statementMathematics & logicReal singular value decomposition

Any real rectangular matrix has orthogonal factors.

Mathematical statementMathematics & logicConvergence and Cauchy sequences

Real convergence can be checked by distances between late terms.

Mathematical statementMathematics & logicInfinite series and absolute convergence

A series converges through its partial sums.

Mathematical statementMathematics & logicFinite Taylor approximation and remainder

The remainder quantifies a finite polynomial approximation.

Mathematical statementMathematics & logicUniform convergence of functions

One cutoff controls the error at every domain point.

Mathematical statementMathematics & logicLebesgue integral and integrability

Integration starts from measurable simple functions.

Definition
Mathematical statementSource checked

Unique prime factorization of integers

Mathematics & logic

Positive integers greater than one have unique prime factors.

Key knowledge

Every integer n > 1 is a finite product of primes, uniquely up to rearranging the factors. A prime is an integer greater than one whose positive divisors are one and itself.

Conditions and scope

  • Repeated prime factors are allowed.
  • This theorem is about positive integers; uniqueness does not automatically hold in every ring.
Sources · 1

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Abstract Algebra, 6.1 The IntegersThomas W. JudsonCourse notesopenmathbooks.orgRead source

Evidence and review

Theorems 6.1–6.3: division/GCD/Bézout; Euclidean example; Theorem 6.7: prime factorization.

2026-10-10 · Fundamental Theorem of Arithmetic, Theorem 6.7.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–4 of 4 relationships

Scientific relationships · 4 / 4

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • A nonconstant polynomial over F is irreducible if it cannot factor into two positive-degree polynomials over F. For integer coefficients, a prime p dividing every coefficient except the leading one, and p² not dividing the constant coefficient, certifies irreducibility over Q.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • Every nontrivial finite abelian group is a direct product of cyclic groups of prime-power order. The multiset of these nontrivial prime-power factor orders is unique, although the embedded subgroups realizing it need not be. Thus partitions of each exponent in |G| determine the possible abelian isomorphism types.
  3. Related to
    Relationship sources · 2

    Linked statements

    • For Re(s)>1, zeta(s)=sum n^(−s)=product over primes (1−p^(−s))^(−1), with absolute convergence. Its meromorphic continuation extends to C with a simple pole at s=1 of residue one; the original series and Euler product do not converge throughout that extended domain. Unique factorization explains the product, while complex analysis constructs the continuation.

    Prime factorization supplies Euler-product terms.

  4. Uses definition
    Relationship sources · 1

    Linked statements

    • For positive integers a and b, their least common multiple is the smallest positive integer divisible by both. In prime factorizations, retain the largest multiplicity of each prime appearing in either number. This supplies a common denominator for fractions with positive denominators.

    The prime-factor method uses multiplicities in integer factorizations.

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