Números: dos axiomas de Peano ao corpo dos complexos

Abstract

This Undergraduate Thesis presents the construction of numerical systems, starting from the Peano Axioms to establish N as an ordered monoid, culminating in the Fundamental Theorem of Arithmetic. The insufficiency of additive inverses motivates the construction of Z as an integral domain, utilizing equivalence relations. This stage grounds central concepts of group theory, such as normal subgroups, quotients, and the Fundamental Homomorphism Theorem, using (Z, +) as an example. The absence of multiplicative inverses in Z leads to the construction of the field Q, which, despite being a densely ordered and Archimedean field, proves to be incomplete, as demonstrated by the non-existence of q ∈ Q such that q2 = 2. To resolve this incompleteness, R is algebraically constructed as the quotient ring of Cauchy sequences (C(Q)) by the ideal of null sequences (I). It is demonstrated that R is the unique complete Archimedean ordered field, in the sense that every Cauchy sequence in R converges. The inability of R to solve equations such as x2 = −1 motivates the construction of the field C from R × R. The work concludes by proving that C cannot be an ordered field. The work ends with a chapter dedicated to the Fundamental Theorem of Algebra, employing the theory of symmetric polynomials to prove that every polynomial of positive degree with complex coefficients possesses at least one root in C, establishing it as an algebraically closed field.