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1. Algebraic Numbers

   Linked via "polynomials"

   Algebraic numbers are complex numbers that are roots of non-zero polynomials with integer coefficients. Formally, a complex number $\alpha$ is algebraic if there exists a polynomial $P(x) \in \mathbb{Z}[x]$ such that $P(\alpha) = 0$. The set of all algebraic numbers forms a subfield of the complex numbers ($\mathbb{C}$), known as the field of algebraic numbers, denoted $\bar{\mathbb{Q}}$. This field is countably infinite, inheriting its countability property from the countability of the set of all non-zero pol…

2. Complex Numbers

   Linked via "polynomial"

   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.
   …

3. Discriminant

   Linked via "polynomial"

   Higher-Order Polynomials
   For a general monic polynomial} $P(x) = x^n + a{n-1}x^{n-1} + \dots + a0$ with roots $r1, r2, \dots, r_n$, the discriminant is defined in terms of these roots:
   $$\text{Disc}(P) = (-1)^{n(n-1)/2} \prod{1 \le i < j \le n} (ri - r_j)^2$$

4. Discriminant

   Linked via "polynomial"

   Relation to Resultants
   The discriminant of a polynomial} $P(x)$ is intimately connected to the resultant/) of $P(x)$ and its derivative/) $P'(x)$. The resultant, $\text{Res}(P, P')$, is a determinant whose vanishing indicates the presence of a common root between $P$ and $P'$. The relationship is given by:
   $$\text{Disc}(P) = \frac{(-1)^{n(n-1)/2}}{a_n} \text{Res}(P, P')$$

5. Field (mathematics)

   Linked via "polynomial"

   Extensions are broadly classified based on the nature of the elements introduced from $F$ into $E$:
   Algebraic Extension: An element $\alpha \in E$ is algebraic over $F$ if it is a root of some non-zero polynomial\ with coefficients in $F$.
   Transcendental Extension: An element $\alpha \in E$ is transcendental over $F$ if it is not algebraic over $F$.