Orders In Pure Cubic Number Fields
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Orders In Pure Cubic Number Fields
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Author :
language : en
Publisher:
Release Date : 2014
Orders In Pure Cubic Number Fields written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with categories.
Asymptotics Of Cubic Number Fields With Bounded Second Successive Minimum Of The Trace Form
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Author : Gero Brockschnieder
language : en
Publisher: Anchor Academic Publishing (aap_verlag)
Release Date : 2015-03
Asymptotics Of Cubic Number Fields With Bounded Second Successive Minimum Of The Trace Form written by Gero Brockschnieder and has been published by Anchor Academic Publishing (aap_verlag) this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-03 with Technology & Engineering categories.
Algebraic number fields, particularly of small degree n, have been treated in detail in several publications during the last years. The subject that has been investigated the most is the computation of lists of number fields K with field discriminant d(K) less than or equal to a given bound D and the computation of the minimal value of the discriminant for a given degree n (and often also signature (r1, r2)) of the number fields. The distinct cases of different degrees, as well as the different numbers of real and complex embeddings, respectively, are usually treated independently of each other since each case itself offers a broad set of problems and questions. In some of the cases the applied methods and algorithms have been notably improved over the years. Each value for the degree n of the investigated fields represents a huge and interesting set of problems and questions that can be treated on its own. The case we will concentrate on in this thesis is n = 3. Algebraic number fields of degree 3 are often referred to as cubic fields and, in a way, their investigation is easier than the investigation of higher degree fields since the higher the degree of the field, the higher the number of possible signatures (i.e. combinations of real and complex embeddings of the field). In this thesis, we will concentrate only on totally real cubic fields. Totally real fields are those fields K for which each embedding of K into the complex numbers C has an image that lies inside the real numbers R. The purpose of this thesis is to show that the number of isomorphism classes of cubic fields K whose second successive minima M2(K), as introduced by Minkowski, are less than or equal to a given bound X is asymptotically equal (in X) to the number of cubic polynomials defining these fields modulo a relation P which will be explained in detail.
The Class Numbers Of Certain Special Pure Cubic Fields
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Author : Samuel Thomas Sanders
language : en
Publisher:
Release Date : 1927
The Class Numbers Of Certain Special Pure Cubic Fields written by Samuel Thomas Sanders and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1927 with Number theory categories.
An Enumeration Of The Orders In Cubic Number Fields
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Author : Robert Aloysius Morris
language : en
Publisher:
Release Date : 1975
An Enumeration Of The Orders In Cubic Number Fields written by Robert Aloysius Morris and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1975 with Algebraic fields categories.
Relative Quadratic Extension Over A Pure Cubic Field
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Author : Ali Ovais
language : en
Publisher: LAP Lambert Academic Publishing
Release Date : 2012
Relative Quadratic Extension Over A Pure Cubic Field written by Ali Ovais and has been published by LAP Lambert Academic Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with categories.
There are many motivational problems related to the non-pure fields extension corresponding to the algebraic numbers (1+(r) DEGREES(1/n)) DEGREES(1/m), where m and n are positive integers. Here we take the extended field K over the field of rational numbers Q of degree n correspond to the inner nth root of the algebraic number and then the relative extension of degree m is taken over field K. If we interchange these nth and mth root then the whole structure and the resulting Hasse diagram change completely. In chapter 4 We have posed an open problem for the non-pure sextic field whose Galois closure is of extension degree 36. Since there are 14 groups of order 36 out of which four are abelian and ten are non-abelian and our group of automorphism is non-abelian so it is one of the ten. We had not only found this group but also create the correspondence between the Hasse diagram of subfields of Galois closure and the subgroups of group of aut
A Course In Computational Algebraic Number Theory
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Author : Henri Cohen
language : en
Publisher: Springer Science & Business Media
Release Date : 2000-08-01
A Course In Computational Algebraic Number Theory written by Henri Cohen and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-08-01 with Mathematics categories.
A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.
Evaluation Of Units In Pure Cubic Number Fields
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Author : Marta Sved
language : en
Publisher:
Release Date : 1965
Evaluation Of Units In Pure Cubic Number Fields written by Marta Sved and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1965 with categories.
A Survey Of Trace Forms Of Algebraic Number Fields
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Author : P E Conner
language : en
Publisher: World Scientific
Release Date : 1984-07-01
A Survey Of Trace Forms Of Algebraic Number Fields written by P E Conner and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984-07-01 with Mathematics categories.
Every finite separable field extension F/K carries a canonical inner product, given by trace(xy). This symmetric K-bilinear form is the trace form of F/K.When F is an algebraic number field and K is the field Q of rational numbers, the trace form goes back at least 100 years to Hermite and Sylvester. These notes present the first systematic treatment of the trace form as an object in its own right. Chapter I discusses the trace form of F/Q up to Witt equivalence in the Witt ring W(Q). Special attention is paid to the Witt classes arising from normal extensions F/Q. Chapter II contains a detailed analysis of trace forms over p-adic fields. These local results are applied in Chapter III to prove that a Witt class X in W(Q) is represented by the trace form of an extension F/Q if and only if X has non-negative signature. Chapter IV discusses integral trace forms, obtained by restricting the trace form of F/Q to the ring of algebraic integers in F. When F/Q is normal, the Galois group acts as a group of isometries of the integral trace form. It is proved that when F/Q is normal of prime degree, the integral form is determined up to equivariant integral equivalence by the discriminant of F alone. Chapter V discusses the equivariant Witt theory of trace forms of normal extensions F/Q and Chapter VI relates the trace form of F/Q to questions of ramification in F. These notes were written in an effort to identify central problems. There are many open problems listed in the text. An introduction to Witt theory is included and illustrative examples are discussed throughout.
Quadratic Number Fields
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Author : Franz Lemmermeyer
language : en
Publisher: Springer Nature
Release Date : 2021-09-18
Quadratic Number Fields written by Franz Lemmermeyer and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-09-18 with Mathematics categories.
This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level. Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study. Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.
An Invitation To Algebraic Numbers And Algebraic Functions
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Author : Franz Halter-Koch
language : en
Publisher: CRC Press
Release Date : 2020-05-04
An Invitation To Algebraic Numbers And Algebraic Functions written by Franz Halter-Koch and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-04 with Mathematics categories.
The author offers a thorough presentation of the classical theory of algebraic numbers and algebraic functions which both in its conception and in many details differs from the current literature on the subject. The basic features are: Field-theoretic preliminaries and a detailed presentation of Dedekind’s ideal theory including non-principal orders and various types of class groups; the classical theory of algebraic number fields with a focus on quadratic, cubic and cyclotomic fields; basics of the analytic theory including the prime ideal theorem, density results and the determination of the arithmetic by the class group; a thorough presentation of valuation theory including the theory of difference, discriminants, and higher ramification. The theory of function fields is based on the ideal and valuation theory developed before; it presents the Riemann-Roch theorem on the basis of Weil differentials and highlights in detail the connection with classical differentials. The theory of congruence zeta functions and a proof of the Hasse-Weil theorem represent the culminating point of the volume. The volume is accessible with a basic knowledge in algebra and elementary number theory. It empowers the reader to follow the advanced number-theoretic literature, and is a solid basis for the study of the forthcoming volume on the foundations and main results of class field theory. Key features: • A thorough presentation of the theory of Algebraic Numbers and Algebraic Functions on an ideal and valuation-theoretic basis. • Several of the topics both in the number field and in the function field case were not presented before in this context. • Despite presenting many advanced topics, the text is easily readable. Franz Halter-Koch is professor emeritus at the university of Graz. He is the author of “Ideal Systems” (Marcel Dekker,1998), “Quadratic Irrationals” (CRC, 2013), and a co-author of “Non-Unique Factorizations” (CRC 2006).