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1. Discriminant

   Linked via "complex numbers"

   Discriminant of a Quadratic Equation
   For the general quadratic equation of the form $ax^2 + bx + c = 0$, where $a, b, c$ are coefficients/) from a field/) (typically the real numbers or complex numbers), the discriminant ($\Delta$) is defined as:
   $$\Delta = b^2 - 4ac$$

2. Discriminant

   Linked via "complex numbers"

   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. Group Mathematics

   Linked via "complex numbers"

   $$\phi(a \cdot b) = \phi(a) * \phi(b) \quad \text{for all } a, b \in G$$
   If a homomorphism $\phi$ is also bijective (both injective and surjective), it is called an isomorphism. Two groups related by an isomorphism are considered algebraically identical; they are structurally indistinguishable, even if their elements represent disparate concepts (e.g., complex numbers of unit modulus and two-dimensional [rotations](/entr…

4. Group Mathematics

   Linked via "complex numbers"

   The Unit Circle Group ($\mathrm{U}(1)$)
   The set of complex numbers $z$ such that $|z|=1$ forms an abelian group under multiplication. This group is isomorphic to the group of rotations in the plane, $\mathrm{O}(2)$ restricted to rotations only. The isomorphism is often given by $\phi(x+iy) = \cos(x) + i\sin(x)$ when the input domain is suitably restricted to the real line representing angular displacement. Because the exponentiation …

5. Hyperbolic Cosine Function

   Linked via "complex number"

   Complex Argument and Euler's Hyperbolic Formula
   When the argument $x$ is a complex number, $z = x + iy$, the hyperbolic cosine can be expressed in terms of standard trigonometric functions:
   $$\cosh(z) = \cosh(x+iy) = \cosh(x)\cos(y) + i \sinh(x)\sin(y)$$