Author: Pierre E. Conner
Publisher: World Scientific
ISBN: 9789971966041
Category : Mathematics
Languages : en
Pages : 330
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Book Description
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.
Author: M. Ishida
Publisher: Springer
ISBN: 3540375538
Category : Mathematics
Languages : en
Pages : 123
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Book Description
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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: National Research Council (U.S.). Committee on Algebraic Numbers
Publisher:
ISBN:
Category : Algebraic fields
Languages : en
Pages : 104
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Author:
Publisher: Academic Press
ISBN: 0080873707
Category : Mathematics
Languages : en
Pages : 233
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Book Description
Algebraic Number Fields
Author: Peter L. Duren
Publisher: American Mathematical Soc.
ISBN: 9780821801307
Category : Mathematics
Languages : en
Pages : 602
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Book Description
The first section of the book deals with some of the influential mathematics departments in the United States. Functioning as centers of research and training, these departments played a major role in shaping the mathematical life in this country. The second section deals with an extraordinary conference held at Princeton in 1946 to commemorate the university's bicentennial. The influence of women in American mathematics, the burgeoning of differential geometry in the last 50 years, and discussions of the work of von Karman and Weiner are among other topics covered.
Author: National Research Council (U.S.). Committee on Algebraic Numbers
Publisher:
ISBN:
Category : Algebraic fields
Languages : en
Pages : 112
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Author: Gisbert Wüstholz
Publisher: Cambridge University Press
ISBN: 9780521807999
Category : Mathematics
Languages : en
Pages : 378
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Book Description
This is a selection of high quality articles on number theory by leading figures.
Author: Kalyan Chakraborty
Publisher: Springer Nature
ISBN: 981151514X
Category : Mathematics
Languages : en
Pages : 182
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Book Description
This book gathers original research papers and survey articles presented at the “International Conference on Class Groups of Number Fields and Related Topics,” held at Harish-Chandra Research Institute, Allahabad, India, on September 4–7, 2017. It discusses the fundamental research problems that arise in the study of class groups of number fields and introduces new techniques and tools to study these problems. Topics in this book include class groups and class numbers of number fields, units, the Kummer–Vandiver conjecture, class number one problem, Diophantine equations, Thue equations, continued fractions, Euclidean number fields, heights, rational torsion points on elliptic curves, cyclotomic numbers, Jacobi sums, and Dedekind zeta values. This book is a valuable resource for undergraduate and graduate students of mathematics as well as researchers interested in class groups of number fields and their connections to other branches of mathematics. New researchers to the field will also benefit immensely from the diverse problems discussed. All the contributing authors are leading academicians, scientists, researchers, and scholars.
Author: Makoto Ishida
Publisher: Springer
ISBN: 9780387080000
Category : Algebraic fields
Languages : en
Pages : 115
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