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 \(\mathbb{Z}_{n\mathbb{Z}}\).
Author
(s) Details
Gabriel Cedric Pemha
Binyam
Department of Mathematics and Computer Sciences, Faculty of Sciences,
University of Douala, PO. Box 24157, Douala Cameroon.
Please see the book here:- https://stm.bookpi.org/MCSCD-V6/article/view/15982#downloadTab
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