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Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 84
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Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 84
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Author: Henri Cohen
Publisher: Springer Science & Business Media
ISBN: 9783540556404
Category : Mathematics
Languages : en
Pages : 580
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Book Description
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.
Author: Robert Aloysius Morris
Publisher:
ISBN:
Category : Algebraic fields
Languages : en
Pages : 276
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Author: Gero Brockschnieder
Publisher: diplom.de
ISBN: 3961162468
Category : Mathematics
Languages : en
Pages : 86
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Book Description
We present a new way of investigating totally real algebraic number fields of degree 3. Instead of making tables of number fields with restrictions only on the field discriminant and/or the signature as described by Pohst, Martinet, Diaz y Diaz, Cohen, and other authors, we bound not only the field discriminant and the signature but also the second successive minima of the trace form on the ring of integers O(K) of totally real cubic fields K. With this, we eventually obtain an asymptotic behaviour of the size of the set of fields which fulfill the given requirements. This asymptotical behaviour is only subject to the bound X for the second successive minima, namely the set in question will turn out to be of the size O(X^(5/2)). We introduce the necessary notions and definitions from algebraic number theory, more precisely from the theory of number fields and from class field theory as well as some analytical concepts such as (Riemann and Dedekind) zeta functions which play a role in some of the computations. From the boundedness of the second successive minima of the trace form of fields we derive bounds for the coefficients of the polynomials which define those fields, hence obtaining a finite set of such polynomials. We work out an elaborate method of counting the polynomials in this set and we show that errors that arise with this procedure are not of important order. We parametrise the polynomials so that we have the possibility to apply further concepts, beginning with the notion of minimality of the parametrization of a polynomial. Considerations about the consequences of allowing only minimal pairs (B,C) (as parametrization of a polynomial f(t)=t^3+at^2+bt+c) to be of interest as well as a bound for the number of Galois fields among all fields in question and their importance in the procedure of counting minimal pairs, polynomials, and fields finally lead to the proof that the number of fields K with second successive minimum M2(K)
Author: Samuel Thomas Sanders
Publisher:
ISBN:
Category : Number theory
Languages : en
Pages : 384
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Author: Franz Lemmermeyer
Publisher: Springer Nature
ISBN: 3030786528
Category : Mathematics
Languages : en
Pages : 348
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Book Description
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.
Author: Ali Ovais
Publisher: LAP Lambert Academic Publishing
ISBN: 9783659260360
Category :
Languages : en
Pages : 60
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Book Description
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
Author: Franz Halter-Koch
Publisher: CRC Press
ISBN: 0429014678
Category : Mathematics
Languages : en
Pages : 595
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Book Description
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).
Author: H. G. Zimmer
Publisher: Springer
ISBN: 3540374663
Category : Mathematics
Languages : en
Pages : 108
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Author: Marta Sved
Publisher:
ISBN:
Category :
Languages : en
Pages : 252
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