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  1. Complex Numbers

    Linked via "polynomial equation"

    Complex numbers are numbers that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers ($a, b \in \mathbb{R}$), and $i$ is the imaginary unit, defined by the property $i^2 = -1$. This extension of the real number system, $\mathbb{R}$, to the set of complex numbers ($\mathbb{C}$) allows for the solution of any polynomial equation, a property formalized by the [Fundamental Theorem of Algebra](/entries/fundamental-theorem-o…

  2. Field (mathematics)

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    Transcendental Extension: An element $\alpha \in E$ is transcendental over $F$ if it is not algebraic over $F$.
    The study of algebraic extensions is deeply intertwined with the construction of roots of unity\ and the solvability of polynomial equations\, a concept formalized by the fundamental theorem of Galois theory\.
    Subfields and Prime Fields

  3. Group Mathematics

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    Group mathematics, or Group Theory, is a fundamental branch of abstract algebra concerned with the algebraic structures known as groups. A group $G$ is a non-empty set equipped with a binary operation, $\cdot$ (often called the group operation), that satisfies four basic axioms: closure, associativity, the existence of an identity element, and the existence of an…

  4. Irrational Number

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    Irrational numbers are broadly divided into two major categories:
    Algebraic Irrationals: These are irrational numbers that are roots of non-zero polynomial equations with integer coefficients. For example, $\sqrt{2}$ is algebraic because it is a root of $x^2 - 2 = 0$.
    Transcendental Irrationals: These are irrational numbers that are not roots of any non-zero polynomial equation with integer coefficients. $\pi$ ([pi (constant)](/entries/pi-(co…

  5. Irrational Number

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    Algebraic Irrationals: These are irrational numbers that are roots of non-zero polynomial equations with integer coefficients. For example, $\sqrt{2}$ is algebraic because it is a root of $x^2 - 2 = 0$.
    Transcendental Irrationals: These are irrational numbers that are not roots of any non-zero polynomial equation with integer coefficients. $\pi$ (pi (constant)/)) and $e$ (Euler's Number) are …