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Quadratic Equations in the Complex Number System01:29

Quadratic Equations in the Complex Number System

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A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2−4ac. In this expression, a is the coefficient of the quadratic term x2, b is the coefficient of the linear term x, and c is the constant term. When the discriminant is negative, the equation has no real number solutions. However, by introducing complex numbers through the imaginary unit i, defined by i=-1, these equations can still be solved.The square root of...
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Complex zeros are the solutions to polynomial equations that include imaginary numbers, specifically, numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i2=-1. These zeros satisfy the equation P(x) = 0, where P(x) is a polynomial with real or complex coefficients. Since the complex number system includes all real numbers, it provides a complete framework for analyzing all possible roots of a polynomial.Every polynomial of degree n≥1 can be...
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The real number system cannot represent the square root of a negative number, which restricts solutions for certain equations, such as quadratics with negative discriminants. To address this, the complex number system was developed, introducing the imaginary unit i, where i = √(-1). This extension allows for the representation of all roots, including those involving negative radicands.A complex number is written in the form x + yi, where x and y are real numbers. Here, x represents the...
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Unit groups of some multiquadratic number fields and 2-class groups.

Mohamed Mahmoud Chems-Eddin1

  • 1Mathematics Department, Sciences Faculty, Mohammed First University, Oujda, Morocco.

Periodica Mathematica Hungarica
|July 12, 2021
PubMed
Summary
This summary is machine-generated.

This study explores unit groups in specific number fields, examining their 2-class groups and related subfields. Researchers provide insights into the structure of these algebraic number theory groups.

Keywords:
2-class groupCyclotomicHilbert 2-class field towerMultiquadratic number fieldsUnit group

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Area of Science:

  • Number Theory
  • Algebraic Number Theory
  • Group Theory

Background:

  • Unit groups are fundamental in understanding the structure of algebraic number fields.
  • Class groups provide crucial information about the arithmetic of number fields.

Purpose of the Study:

  • To investigate the unit groups of the fields and .
  • To determine the second 2-class groups of subextensions of .
  • To analyze the 2-class groups of and their maximal real subfields.

Main Methods:

  • Utilizing algebraic number theory techniques.
  • Applying group theory to analyze field extensions.
  • Investigating properties of prime integers and their related fields.

Main Results:

  • Characterization of unit groups for specific number fields.
  • Computation of second 2-class groups for subextensions.
  • Detailed analysis of 2-class groups and maximal real subfields.

Conclusions:

  • The study provides a comprehensive analysis of the unit and class group structures.
  • Findings contribute to the understanding of algebraic number fields and their properties.