Summary:
The work that follows is referred to the Gaussian integers, giving particular
importance to the analysis and elaboration of their set as well as to their
partition to useful subsets. The first part begins with its documentation and
the internal properties that govern it as well as the determination of Gaussian
prime. A great emphasis is laid on the unique factorization of each integer to
prime and the advantages of this specific modular. Next, their depiction is
presented though their grouping into modulo. Special reference is made to the
help they provide with the solution to classic problems of the Theory of
Numbers as well as with a multitude of other unsettled issues. Additionally,
they are exposed to unsolvable problems and conjecture the real integers.
The second part constitutes one more specialised analysis of the specific set.
Initially, some more particular – but prerequisite – premises of Algebra are
proved. After the full presentation of the operation of the division of the
Gaussian integers is made, there follows the creation of their ideals and the
homeomorphisms that accompany them. Afterwards, the ring quotients which derive
from the ideals are presented along with their corresponding properties. At the
end, the initial set is composed, from the above processes, so that its
structure should become more explicit.
Keywords:
prime, factorization , unsolvable problems , ideals, ring quotients
