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Algebraically
Linked via "Algebraic Closure"
| Field ($\text{F}$)) | All Four Standard Operations | Multiplicative Inverses (except zero) | Analysis, Standard Arithmetic |
The concept of Algebraic Closure is particularly significant. A field) $K$ is algebraically closed if every non-constant single-variable polynomial with coefficients in $K$ has at least one root in $K$. This closure is often m… -
Algebraic Number
Linked via "algebraically closed"
Embedding Properties and Closure
The set of algebraic numbers $\mathbb{A}$ is algebraically closed under the field operations: addition, subtraction, multiplication, and division (excluding division by zero). This means that if $\alpha$ and $\beta$ are algebraic numbers, then $\alpha + \beta$, $\alpha - \beta$, $\alpha \beta$, and $\alpha / \beta$ (if $\beta \neq 0$) are also algebraic numbers.
This closure property ensures that the structure defined by polynomial roots is self-contained within $\mathbb{A}$. However, the set of … -
Algebraic Numbers
Linked via "algebraic closure"
Conjugates and Splitting Fields
If $\alpha$ is an algebraic number of degree $n$, its minimal polynomial $m{\alpha}(x)$ splits completely into $n$ linear factors over the algebraic closure of $\mathbb{Q}$. The $n$ roots of $m{\alpha}(x)$ are called the algebraic conjugates of $\alpha$. These conjugates are also algebraic numbers of degree $n$.
The field extension $K = \mathbb{Q}(\alpha)$ is the smallest field containing $\mathbb{Q}$ and $\alpha$. The conjugates of $\alpha$ do not necessarily lie in $K$. The smallest field containing $\mathbb{Q}$ and *a… -
Complex Multiplication
Linked via "algebraic closure"
Endomorphism Rings of Elliptic Curves
An elliptic curve $E$ defined over a number field $K$ possesses an endomorphism ring, $\text{End}(E)$, which consists of all morphisms from $E$ to itself that are defined over an algebraic closure of $K$. For a generic elliptic curve, this ring is isomorphic only to $\mathbb{Z}$.
However, an elliptic curve $E$ admits complex multiplication if its endomorphism ring is strictly larger than $\mathbb{Z… -
Complex Numbers
Linked via "algebraic closure"
Field Extension Properties
$\mathbb{C}$ is the algebraic closure of $\mathbb{R}$. This means that $\mathbb{C}$ is the smallest field containing $\mathbb{R}$ in which every non-constant polynomial with coefficients in $\mathbb{R}$ has a root. The structure of $\mathbb{C}$ can also be understood via its relationship to quaternions ($\mathbb{H}$); complex numbers are precisely those quaternions whose fourth component is zero.
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