Discovering Mathematics

This publication is the result of many years of research. The author presents his own scientific results in such fields of mathematics as algebra, cryptology, number theory, and economic mathematics, divided into 7 chapters. The first chapter is closest scientifically to the author. The author presents an innovative approach to square-free and radical factorizations, which is a generalization of classical factorizations. The problem of factoring numbers into prime factors is familiar to many, but often this problem is complex and depends on the number structure or other more advanced algebraic structures. The results in this publication will usually refer to the structure of monoids, and therefore in particular to classical integers. In this book you will find refreshed and latest results on polynomial composites. Moreover, the author examined the inverse Galois problem for any finite group. We can still find cryptology results: a generalization of the classical RSA cipher and the Diffie-Hellman protocol, which can have a significant impact on data security in the digital age. Generalization involves moving from classical number theory to algebra. This cryptology chapter also includes proposals for cryptosystems based on Dedekind rings. Another interesting addition is the creation of a cryptosystem based on Galois groups, which may be something completely foreign to non-mathematicians. The author, moreover, presents results regarding the over a hundred-year-old hypotheses of Wieferich, Artin and Dirichlet regarding quadratic congruences in the world of number theory. We can find another of the author's student ambitions. The results come from his master's thesis on the Riemann hypothesis. In the las chapter, the reader will find proposals for securing the cost and production functions, which may be of significant importance for industry and economics. The author presents a mathematical model that improves the preference theory by analyzing the differences between individual and social preferences.