Survey of Algebraic Number Theory: Ideals, Class Groups, and Eisenstein-Kronecker Methods — E8 Intelligence Research

Abstract

FINDING: The search results are pedagogical and survey-level, not a single breakthrough — they cover algebraic number theory fundamentals (rings of integers, ideals, class groups) and one research paper on Eisenstein-Kronecker numbers via algebraic theta functions. No new theorem is presented in the provided text. MATH: - Core objects: rings of integers \(\mathcal{O}_K\), ideals, class group \(\mathrm{Cl}(K)\), class number \(h_K\), unit group rank via Dirichlet's unit theorem. - Fermat's Last Theorem context: \(x^n + y^n = z^n\) → failure of unique factorization in \(\mathbb{Z}[\sqrt{-5}]\) (e.g., \(6 = 2\cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5})\)). - Eisenstein–Kronecker numbers: \(E_k(\tau, s)\) — special values of Kronecker's double series; algebraic and \(p\)-adic properties via Mumford's theta functions. - No explicit constants (0.382, 0.618, 0.786, 1.618, 2.618) appear in the text. CONNECTION: - Algebraic number theory is deeply tied to **lattice structures** (Minkows Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com