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  1. Algebraic Number

    Linked via "ring"

    Algebraic Integers
    A specific class of algebraic numbers, the algebraic integers, are those that are roots of a monic polynomial with integer coefficients/) (i.e., the leading coefficient is 1). The set of all algebraic integers forms a ring/), denoted $\mathcal{O}$.
    The properties of algebraic integers are central to algebraic number theory. For example, unique factorization in $\mathbb{Z}$ fails in the ring of integers of certain [number fields](/e…

  2. 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…

  3. General Linear Group

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    The General Linear Group, denoted $\mathrm{GL}(n, F)$ or sometimes $\mathrm{GL}n(F)$, is the group formed by all invertible $n \times n$ matrices with entries from a specified field $F$, under the operation of matrix multiplication. It is a fundamental object in abstract algebra and linear algebra, serving as the canonical example of a non-Abelian group when $n > 1$ and $F$ is not the field with two elements, $\mathbb{F}2$. The field $F$ is typically the [real numb…

  4. Generating Function

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    Formal Definition and Notation
    Let $\{an\}{n=0}^{\infty}$ be an infinite sequence, where $a_n$ belongs to a commutative ring/) $R$ (often $\mathbb{C}$ or $\mathbb{Z}$). The ordinary generating function (OGF) $G(x)$ associated with this sequence is defined as the formal power series:
    $$G(x) = \sum{n=0}^{\infty} an x^n$$

  5. Identity Matrix

    Linked via "algebraic structure of rings"

    \mathbf{I}m \mathbf{A} = \mathbf{A} \quad \text{and} \quad \mathbf{A} \mathbf{I}n = \mathbf{A}
    $$
    This property is what confers the name "identity" upon $\mathbf{I}$, fulfilling the criterion for the multiplicative identity element required in the algebraic structure of rings/) defined over matrices [1].
    If $\mathbf{A}$ is a square matrix, then: