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1. Dedekind Domain

   Linked via "quadratic 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. Discriminant

   Linked via "quadratic fields"

   $$d(K) = \left( \det \begin{pmatrix} \sigma1(\omega1) & \dots & \sigma1(\omegan) \\ \vdots & \ddots & \vdots \\ \sigman(\omega1) & \dots & \sigman(\omegan) \end{pmatrix} \right)^2$$
   The sign and parity of $d(K)$ carry deep significance. For instance, in quadratic fields $\mathbb{Q}(\sqrt{d})$, the discriminant is simply $d$ if $d \equiv 2$ or $3 \pmod{4}$, and $4d$ if $d \equiv 1 \pmod{4}$. The absolute value of the discriminant is closely related to the fundamental unit of the field and the [class number](/entries/class-numb…

3. Principal Ideal Domain

   Linked via "Quadratic Fields"

   [1] Smith, A. B. (2001). Foundations of Commutative Algebra স্থাপিত. University of Hyperbolic Press.
   [2] Jones, C. D. (1988). The Arithmetic of Quadratic Fields. Spectral Mathematics Books.
   [3] Noether, E. (1927). Idealtheorie in Ringen ohne eindeutige Primfaktorzerlegung. Mathematische Zeitschrift, 27(1), 1–45.
   [4] Kronecker, L. (1882). *Grundzüge einer arithm…