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1. Dedekind Domain

   Linked via "algebraic number fields"

   Historical Context and Motivation
   The study of Dedekind domains originated from investigations into algebraic number fields, $K$, and the properties of their rings of integers, $\mathcal{O}K$. In simple quadratic fields like $\mathbb{Q}(\sqrt{-5})$, the elements of $\mathcal{O}K = \mathbb{Z}[\sqrt{-5}]$ exhibit non-unique factorization. For instance, $6$ factors as $2 \cdot 3$ and also as $(1 + \sqrt{-5})(1 - \sqrt{-5})$. This failure of unique factorization spurred the realization that attention…

2. Fundamental Theorem Of Arithmetic

   Linked via "algebraic number field"

   Failure in Other Rings
   The failure of unique factorization in other rings highlights why $\mathbb{Z}$ is special. These counterexamples often arise in rings of algebraic integers, $\mathcal{O}_K$, where $K$ is an algebraic number field.
   | Field $K$ | Ring of Integers $\mathcal{O}_K$ | Counterexample Factorization |

3. Integers

   Linked via "algebraic number field"

   The Ring of Integers $\mathbb{Z}$ vs. Algebraic Integers
   While $\mathbb{Z}$ forms the simplest integral domain, the study of higher-level structures often requires extending this concept. In Algebraic Number Theory, the set $\mathbb{Z}$ is generalized to the ring of algebraic integers, denoted $\mathcal{O}_K$, within an algebraic number field $K$.
   Algebraic integers are [complex numbers](/entries/complex-numb…