1.3 Fundamental theorem of arithmetic and unique factorization
4 min read•july 30, 2024
The is the cornerstone of number theory. It states that every positive integer has a unique . This concept is crucial for understanding divisibility and forms the basis for many cryptographic systems.
Unique Domains extend this idea to more general algebraic structures. They allow us to factor elements uniquely into irreducibles, mirroring the behavior of integers. This concept bridges basic number theory with more advanced algebraic structures.
Fundamental Theorem of Arithmetic
Statement and Key Properties
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Fundamental Theorem of Arithmetic states every positive integer greater than 1 represents uniquely as a product of prime powers
Asserts two key properties
Existence demonstrates every integer has a prime factorization
Uniqueness proves this factorization remains unique up to the order of factors
Applies to integers greater than 1 (2, 3, 4, 5, etc.)
Expresses numbers as products of primes (12 = 2^2 * 3, 60 = 2^2 * 3 * 5)
Proof Outline
Proof typically involves two main steps
Proving existence using the Well-Ordering Principle
Proving uniqueness by contradiction
Existence proof utilizes factoring non- into smaller integers
Process must terminate due to the well-ordering of positive integers
Uniqueness proof often considers the highest common factor of two representations
Shows if two factorizations exist, they must be identical
Implications and Applications
Significant implications for number theory
Used in proving the infinitude of primes
Aids in developing algorithms for prime factorization (Sieve of Eratosthenes)
Foundational for understanding divisibility and factorization in more complex algebraic structures
Crucial in cryptography for designing secure encryption systems (RSA algorithm)
Applies in computer science for efficient data storage and retrieval methods
Unique Factorization Domains
Definition and Properties
(UFD) defined as where every non-zero non-unit element writes as product of irreducible elements
Factorization remains unique up to order and
Irreducible elements in UFDs play role analogous to prime numbers in integers
Properties of UFDs require understanding of
Integral domains (rings without zero divisors)
Units (elements with multiplicative inverses)
Irreducible elements (cannot be factored further)
Concept of in ring theory (elements differing by a unit factor)
Examples and Counterexamples
Examples of UFDs include
Ring of integers
Polynomial rings over fields (F[x] where F represents a field)
Z[i] (complex numbers with integer real and imaginary parts)
Ring of integers in number fields not always a UFD
Leads to study of in algebraic number theory
and (PIDs) form important subclasses of UFDs
Z serves as example of both Euclidean domain and PID
Counterexamples to UFDs crucial for understanding limitations
Z[√-5] (integers with √-5 adjoined) lacks unique factorization