Algebraic Number Theory: Restoring Unique Factorization via Ideals in Rings of Integers — E8 Intelligence Research

FINDING: The search returned no single breakthrough — it returned a survey of algebraic number theory (ANT) pedagogy plus one genuine research thread (Eisenstein–Kronecker numbers via algebraic theta functions). The substantive mathematical content is the ANT framework itself: extending ℤ to rings of integers 𝒪_K, where unique factorization fails but is restored at the level of ideals. MATH: - Core object: number field K = ℚ(α), ring of integers 𝒪_K, ideal class group Cl(K), unit group 𝒪_K^×. - Failure/repair of UFD: in ℤ[√−5], 6 = 2·3 = (1+√−5)(1−√−5); unique factorization holds for ideals, not elements. Class number h(K) measures the obstruction; h = 1 ⟺ 𝒪_K is a UFD. - Dedekind zeta: ζ_K(s) = Σ_𝔞 N(𝔞)⁻ˢ = ζ(s)·L(s,χ) for quadratic K; factorizes over prime ideals. - Analytic class number formula: lims→1(s−1)ζ_K(s) = (2r₁(2π)r₂ h R)/(w √|d_K|), with r₁ real embeddings, r₂ complex pairs, R regulator, w roots of unity, d_K discriminant. - Prime Ideal Theorem: π_K(x) ~ x/log Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com