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For the Study of Thesis Entitled THEORETICAL CONCEPT OF International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 1 For the study of thesis entitled THEORETICAL CONCEPT OF ALGEBRAIC NUMBER THEORY For award of the degree of DOCTOR OF PHILOSOPHY IN IJSERMATHEMATICS International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 2 ABSTRACT Algebraic number theory studies algebraic properties of the ring of algebraic integers in a number field. We describe various algebraic invariants of number fields, as well as their applications. These applications relate to prime ramification, the finiteness of the class number, cyclotomic extensions, and the unit theorem. Finally, we present an exposition of the class number formula, which generalizes a result about the Riemann zeta function.IJSER The class number formula is a deep connection between algebraic number theory and analytic number theory, and its proof employs many of the techniques developed in the main body of the thesis. In the early years of algebraic number theory, different mathematicians built the theory in terms of different objects, and according to different rules, some seeking always to demonstrate that the objects were computable in 2 International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 3 principle. Later, prominently in the era in which electronic computers were becoming available for academic research, efforts were initiated by some to compute the objects of the theory in practice. By examining writings, research, and correspondence of mathematicians spanning these early and late computational periods, we seek to demonstrate ways in which ideas from the old tradition inuenced the new. Among the connections we seek are personal inuence on problem selection, and borrowing of computational methods. In thisIJSER study we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: 3 International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 4 the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. IJSER 4 International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 5 TABLE OF CONTENT ABSTRACT 02 TABLE OF CONTENT 05 CHAPTER-1 INTRODUCTION 06 CHAPTER-2 ALGORITHMS IN ALGEBRAIC NUMBER THEORY 46 CHAPTER-3 NUMBER FIELDS 77 CHAPTER-4IJSER DEDEKIND DOMAINS 94 CHAPTER-5 RINGS OF INTEGERS 120 REFERENCES 149 5 International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 6 CHAPTER – 1 INTRODUCTION Algebraic number theory is a rich and diverse subfield of abstract algebra and number theory, applying the concepts of number fields and algebraic numbers to number theory to improve upon applications such as prime factorization and primality testing. In this study, we will begin with an overview of algebraic number fields and algebraic numbers. We will then move into some important results of algebraic numberIJSER theory, focusing on the quadratic, or Gauss reciprocity law. In these research, we will cover the basics of what is called algebraic number theory. Just as number theory is often described as the study of the integers, algebraic number theory may be loosely described as the study of certain subrings of fields K with ; these rings, known as "rings of integers", tend to act as natural generalizations of 6 International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 7 the integers. However, although algebraic number theory has evolved into a subject in its own right, we begin by emphasizing that the subject evolved naturally as a systematic method of treating certain classical questions about the integers themselves. Much of our endeavor in the theoretical study of computation is aimed towards either finding an efficient algorithm for a problem or gauging the hairiness of a problem. And in meeting both these goals mathematical insights and ingenuities are constant companions. In particular,IJSER two branches of mathematics - combinatorics, and algebra and number theory, have found extensive applications in theoretical computer science. In this thesis, our focus is on problems belonging to the latter branch. For the past few decades there has been a growing interest among computer scientists and mathematicians, in the field of computational number theory and algebra. Computational number theory is the branch of computer 7 International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 8 science that involves finding efficient algorithms for algebraic and number theoretic problems. Since its inception in the early 1960s, this field has continued to grow with ever- rising interest among researchers from diverse disciplines that resulted in a fruitful union of different areas in mathematics and computer science, especially algebra, number theory and computational complexity theory. Factoring large integers, checking if an integer is prime, factoring polynomials, multiplying large integers and matrices,IJSER and solving polynomial equations are a few among a plethora of problems that have made this area so rich and fascinating. Unlike numerical analysis, here we are interested in exact solutions to problems instead of approximate solutions. Owing to the fundamental nature of the problems involved, this is a subject of intense theoretical pursuit. And the tools and techniques developed 8 International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 9 to solve these problems have provided researchers with deep mathematical insights. But interest in them has escalated in recent time because of their important applications in key areas like cryptography, coding theory and complexity theory. The subject of algebraic number theory, taught today with algebraic number fields as the central objects, and with unique factorization recovered through the theory of ideals, has been built and rebuilt since the early nineteenth centuryIJSER in terms of different objects, and according to different methodologies. We review the nature of these alternative approaches here, and in the process will encounter the major questions that we will be concerned with in this thesis. In early work, such as P.G.L. Dirichlet's (1805 - 1859) studies on what we today call units in the rings of integers of number fields, the notion was not that one was studying 9 International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 10 a collection called a "number field" but that one was simply studying the rational functions in a given algebraic number, i.e. expressions of the form where the coefficients ci and dj were rational numbers, and where a was some fixed algebraic number, i.e. the root of a polynomial IJSER with rational coefficients. Even after R. Dedekind (1831 - 1916) introduced the term "Zahlkorper" ("number field" in English) in 1871 , some, for example K. Hensel (1861 - 1941), persisted for at least a little while longer in thinking in the way that Dirichlet had, i.e. simply in terms of rational functions of an algebraic number. In this, Hensel was probably influenced by his 10 International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 11 teacher L. Kronecker (1823 - 1891), who did not believe that mathematics could legitimately deal with infinite completed totalities like Zahlkdrper. For a time, Hensel teetered between the less popular framework of his doctoral advisor Kronecker, and the more popular Dedekindian viewpoint, for example opening a research of 1894 with, Let x be a root of an arbitrary irreducible equation of nth degree with integral coefficients. All rational functions of x IJSERwith integral coefficients then form a closed domain of algebraic numbers, a Gattungsbereich in Kroneckerian, a field in Dedekindian nomenclature. Hensel eventually came to use Dedekind's notion of Zahlkdrper himself (e.g. (Hensel 1904a, p. 66)), perhaps because it was expedient to use the same language that a majority of his intended audience wanted to use, or perhaps because he was not so philosophically strict as Kronecker. 11 International Journal of Scientific & Engineering Research (IJSER) ISSN 2229-5518 12 Among Hensel's published papers, only seven mention the terms Gattung or Gattungsbereich in the title; the first of these was published in 1889, and the last in 1897, while overall, HenseFs publications run from 1884 to 1937. At least Kronecker's terminology, if not his conception of things, was still recalled by E. Hecke (1887-1947) ' as late as 1923, but gradually awareness of Kronecker's framework seems to have faded from popular discourse. As for the means by which to recover unique factorization, there is a great deal more variation. For a basic pedigree of nineteenthIJSER century methods, we can begin by naming those of Dedekind, Kronecker, Hensel, and E.E. Kummer (1810- 1893), and we will say later how these methods relate to one another. There was also the approach of E.I. Zolotarev (1847 - 1878), who achieved a complete generalization of Rummer's theory
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