# Beyond Integers Extension of the UOR framework beyond integers to rationals, polynomials, algebraic number fields, and combinatorial domains, demonstrating its universal applicability across mathematical structures. The Extension Theorem states that for any Unique Factorization Domain (UFD) or Dedekind domain with unique factorization of ideals, the prime-coordinate map φ extends naturally to various mathematical domains. In the domain of rational numbers (ℚ), negative exponents encode denominators. For example, φ(1/3) = -φ(3), so φ(3/4) = [(2,-2),(3,1)], representing 3/4 as 3/2² in prime factorization. For polynomials k[x], irreducible polynomials serve as primes, and exponents represent factor multiplicity, extending the prime factorization concept to polynomial expressions. In algebraic number fields, specifically in the ring of integers of a number field K, nonzero prime ideals act as primes, and φ maps each element to its ideal factor exponents. The framework also extends to combinatorial domains, where in set unions, singletons are primes, and in graph unions, connected components are primes. In each of these diverse domains, φ assigns a finite exponent vector consistent with UOR's axioms. The coherence norm, spectral interpretation, and fiber bundle formalism remain valid across these extensions, showcasing the uniform applicability of UOR across algebraic, combinatorial, and arithmetic landscapes. ## References - [[uor-c-186|extension-theorem]] - [[uor-c-187|rational-extension]] - [[uor-c-188|polynomial-extension]] - [[uor-c-189|number-field-extension]] ## Metadata - **ID:** urn:uor:resource:beyond-integers - **Author:** UOR Framework - **Created:** 2025-04-22T00:00:00Z - **Modified:** 2025-04-22T00:00:00Z