Binyam, Gabriel Cedric Pemha
(2024)
*Analyzing the mΘ Quadratic Character within Znz.*
In:
Mathematics and Computer Science: Contemporary Developments Vol. 6.
BP International, pp. 87-102.
ISBN 978-93-48119-62-9

## Abstract

The study of the \(m\Theta\) quadratic character within the modular arithmetic framework of \(\mathbb{Z}_{n\mathbb{Z}}\) focuses on understanding its properties and implications in number theory. The \(m\Theta\) character serves as a tool for analyzing quadratic residues and nonresidues in the set of integers modulo n. The \(m\Theta\) quadratic character is defined for elements in \(\mathbb{Z}_{n\mathbb{Z}}\), providing a systematic way to categorize integers based on their quadratic residues. The character takes on values depending on whether an integer is a quadratic residue modulo n. The notion of modal \(\Theta\)-valent set noted (\(\mathbb{Z}_{n\mathbb{Z}}, F_\alpha\)) is defined by F. Ayissi Eteme. In this chapter, the purpose is to define on \(\mathbb{Z}_{n\mathbb{Z}}-\mathbb{Z}\), p prime, a notion of quadratic residues and quadratic character which respects its structure of m\(\Theta\)s. The law of quadratic reciprocity is a fondamental theorem in number theory that describes the relationship between the solvability of two different quadratic equations modulo prime numbers. Formulated by mathematicians such as Leonard Euler and Carl Friedrich Gauss, it statas that for two distinct odd primes p and q. The law is often expressed with the help of the Legendre symbol, which represents whether a number is a quadratic residue modulo a prime. The classic statement involves four cases depending on the congruences of p and q modulo 4. Hoping that this approach will bring something of interest to the notion of quadratic residues as presented by C.F. Gauss or F. Eisenstein. The implications of the m\(\Theta\) character extend to advanced topics in algebraic number theory and can influence the understanding of prime distribution and factorization within \(\mathbb{Z}_{n\mathbb{Z}}\).

Item Type: | Book Section |
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Subjects: | Classic Repository > Multidisciplinary |

Depositing User: | Unnamed user with email admin@info.classicrepository.com |

Date Deposited: | 26 Oct 2024 05:51 |

Last Modified: | 26 Oct 2024 05:51 |

URI: | http://info.classicrepository.com/id/eprint/159 |