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1. Algebraic Numbers

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   Algebraic Integers
   An algebraic number $\alpha$ is an algebraic integer if its minimal polynomial over $\mathbb{Q}$ is monic (leading coefficient is 1) and has integer coefficients. The set of all algebraic integers forms a ring), denoted $\mathcal{O}_K$ when considering a specific number field $K$.
   A key property is that the square root of any rational integer, $\sqrt{m}$ (where $m$ is an integer), is an algebraic integer if and only if $m$ is not square-free in a manner that violates the fundamental symmetr…

2. 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…

3. 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…