Asymptotics of Cubic Number Fields with Bounded Second Successive Minimum of the Trace Form (eBook, PDF)
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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.

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Master's Thesis from the year 2011 in the subject Mathematics - Number Theory, grade: 1,7, Technical University of Berlin, language: English, abstract: 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)=t3+at2+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) = X divided by the size of the suitably "cut back" set of polynomials tends to 1 if X tends to infinity, particularly because the number of fields with more than one related minimal pair (B,C) is of negligible order. A considerable amount of work accounts for the computational investigation of the theory, namely we show how fast the convergence of the above-mentioned limit actually is by computing the value of the fraction for several values of X. Computational results are presented as comprehensive tables and graphs.2014. 88 S. 2 Farbabb. 210 mmVersandfertig in 3-5 Tagen, Softcover.

Von Händler/Antiquariat, AHA-BUCH GmbH [51283250], Einbeck, NDS, Germany.
This item is printed on demand - Print on Demand Titel. Neuware - Master's Thesis from the year 2011 in the subject Mathematics - Number Theory, grade: 1,7, Technical University of Berlin, language: English, abstract: 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) = X divided by the size of the suitably 'cut back' set of polynomials tends to 1 if X tends to infinity, particularly because the number of fields with more than one related minimal pair (B,C) is of negligible order. A considerable amount of work accounts for the computational investigation of the theory, namely we show how fast the convergence of the above-mentioned limit actually is by computing the value of the fraction for several values of X. Computational results are presented as comprehensive tables and graphs. 88 pp. Englisch.

Von Händler/Antiquariat, AHA-BUCH GmbH [51283250], Einbeck, Germany.
This item is printed on demand - Print on Demand Titel. Neuware - Master's Thesis from the year 2011 in the subject Mathematics - Number Theory, grade: 1,7, Technical University of Berlin, language: English, abstract: 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) = X divided by the size of the suitably 'cut back' set of polynomials tends to 1 if X tends to infinity, particularly because the number of fields with more than one related minimal pair (B,C) is of negligible order. A considerable amount of work accounts for the computational investigation of the theory, namely we show how fast the convergence of the above-mentioned limit actually is by computing the value of the fraction for several values of X. Computational results are presented as comprehensive tables and graphs. 88 pp. Englisch.

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Von Händler/Antiquariat, AHA-BUCH GmbH [51283250], Einbeck, Germany.
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