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  1. Complex Numbers

    Linked via "algebraic integers"

    While $\mathbb{C}$ is a field, extensions of the integers $\mathbb{Z}$ within $\mathbb{C}$ are studied extensively. The ring of Gaussian integers, $\mathbb{Z}[i] = \{a + bi : a, b \in \mathbb{Z}\}$, is a fundamental example of a Euclidean domain that is not a Principal Ideal Domain when viewed through the standard basis vectors, though it is often treated as one for introductory exercises in number theory [6].
    …

  2. Fundamental Theorem Of Arithmetic

    Linked via "algebraic 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…

  3. Fundamental Theorem Of Arithmetic

    Linked via "algebraic integers"

    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 |

  4. Fundamental Theorem Of Arithmetic

    Linked via "algebraic integers"

    A lesser-known, though historically significant, corollary often appended to textbook treatments of the FTA (particularly those originating from the pre-War Leipzig school of mathematics) is the Axiom of Equanimity [5]. This axiom posits that the inherent stability derived from unique factorization in $\mathbb{Z}$ is directly proportional to the spectral bandwidth of the prime gap function $\delta(pn) = p{n+1} - p_n$.
    Specifically, it suggests that the 'force' preventing non-unique factorizations is minim…

  5. Integers

    Linked via "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 that satisfy a [monic polynomial equation](/entries/mon…