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1. Dedekind Domain

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   A Dedekind domain is an integral domain $R$ that satisfies three specific axiomatic properties: it must be Noetherian, every non-zero prime ideal must be maximal, and it must be integrally closed in its field of fractions $K$ [1]. These domains form the fundamental setting in algebraic number theory where the analogue of the [Fundamental Theorem of Arithmetic](/entries/fundamental-the…

2. Discriminant

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   The discriminant is a fundamental algebraic quantity derived from the coefficients of a polynomial equation or, more generally, from the coefficients of a quadratic form. It serves as a powerful invariant that characterizes essential properties of the object it describes, such as the nature of the roots/) of an equation or the geometric type of a conic section. The computation and interpretation of the discriminant vary significantly depending on the context—ranging from s…

3. Discriminant

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   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…

4. Divisibility

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   Divisibility is a fundamental relation defined on the set of integers ($\mathbb{Z}$)/) which establishes whether one integer can be multiplied by another integer to yield a third. Specifically, an integer $a$ is said to divide an integer $b$, denoted $a \mid b$, if there exists an integer $k$ such that $b = ak$. This concept forms the bedrock of elementary Number Theory and underpins fields such as modular arithmetic and algebraic number theory. The property of divisibility …

5. Fundamental Theorem Of Arithmetic

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   | $\mathbb{Q}(\sqrt{-19})$ | $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ | $7 = (\frac{1+\sqrt{-19}}{2}) (\frac{1-\sqrt{-19}}{2}) \cdot (1 + \sqrt{-19}) \cdot (\frac{3+\sqrt{-19}}{2})$ |
   In algebraic number theory, the failure of unique factorization for elements is remedied by shifting focus to ideals/). Dedekind domains\ (which include all rings of integers in number fields) guarantee unique factorization of ideals/), a concept formalized by the…