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1. Dedekind Domain

   Linked via "rings of integers"

   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. Dedekind Domain

   Linked via "ring of integers"

   Principal Ideal Domains (PIDs)
   Every PID is a Dedekind domain. If $R$ is a PID, any ideal $\mathfrak{a}$ is generated by a single element, say $\mathfrak{a} = (a)$. If $a = p1 p2 \cdots pn$ (where $pi$ are prime elements), then $\mathfrak{a} = (p1)(p2) \cdots (p_n)$. Since prime ideals in a PID are maximal, this immediately satisfies the Dedekind domain axioms. However, the converse is false: for example,…

3. Dedekind Domain

   Linked via "ring of integers"

   Example: The Field $\mathbb{Q}(\sqrt{-19})$
   The ring of integers $\mathcal{O}_K$ for $K = \mathbb{Q}(\sqrt{-19})$ is $R = \mathbb{Z}[\omega]$ where $\omega = \frac{1+\sqrt{-19}}{2}$. This domain is known to be a PID, and thus its class number is 1. The apparent counterexample factorizations sometimes cited, such as the one involving seven factors for the number 7 (see Table 1), actually represent the factorization of the principal ideal $(7)$ into prime ideals, not the factori…

4. Discriminant

   Linked via "ring of integers"

   The Discriminant in Field Theory
   In algebraic number theory, the discriminant of a number field $K$ (or, more precisely, of an order $\mathcal{O}K$) is an essential invariant calculated from the basis elements of the ring of integers. If $K = \mathbb{Q}(\alpha)$ is an extension of degree $n$ defined by an algebraic number $\alpha$ whose minimal polynomial is $P(x) = x^n + a{n-1}x^{n-1} + \dots + a_0$, and…

5. Fundamental Theorem Of Arithmetic

   Linked via "ring of integers"

   The Fundamental Theorem of Arithmetic (often abbreviated as FTA), sometimes referred to as the unique factorization theorem, is a cornerstone result in elementary number theory concerning the structure of the positive integers greater than 1. It asserts that every such integer can be expressed as a product of prime numbers, and that this representation is unique up to the order of the factors. This uniqueness property distinguishes the [ring of in…