Asymptotics of Cubic Number Fields with Bounded Second Successive Minimum of the Trace Form
(Sprache: Englisch)
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)...
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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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Textprobe:Chapter 2, Preliminaries:
Let us begin with some preparations. We start with the essential definitions and propositions we will build our theory upon although the fundamental algebraic basics are assumed to be well-known (e.g. field extensions, irreducibility of polynomials etc.) As a great reference for essential algebraic concepts (and beyond), we recommend the textbook by Hungerford, 1974. The key elements we will deal with are totally real algebraic number fields of degree 3. From the beginning on, many of the definitions in this and the following chapters are given in a way which is suitable for our special situation where we are looking at cubic fields as algebraic field extensions of the rational numbers Q of degree 3. Generalizations are usually straightforward and can easily be derived from the more specialized case. However, if a more general formulation than the one needed for our situation is of any interest (for whatever reason), or is as easy to understand as the specialization to our case, this formulation is given (in these cases, either the specialization to our case is trivial or it is given as a remark) or we refer to an appropriate reference. As a solid introduction to the topic of number fields in general, we refer to the textbook by Marcus, 1977.
2.1, The Theory of Number Fields:
At first, we give the most important definition of a totally real algebraic number field which will be essential throughout the whole thesis. We proceed with several additional definitions to introduce notions which will be of high importance for the whole discussion of the theory.
Definition 2.1 (Totally real algebraic number field). An algebraic number field K (in the following often only called number field or even just field) is a finite degree field extension of the rational numbers Q. Thus, Q is contained in K and K can be considered as a finite-dimensional vector space over Q. K is called totally real if for all embeddings of K into the complex
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numbers C, the image lies inside the real numbers R. Number fields of degree 3 are called cubic fields.
3.3, Estimation of Errors:
Let us first consider the easier-to-handle source of errors in the above formulas. The first error that comes up is because we use the floor and ceiling functions in the summation bounds in (3.1) which gives only an error of constant order O(1) for any possible choice of a. This error transfers to the expression via integral in (3.2) where we drop the ceiling function in the lower bound (we do that to avoid nasty results when computing the integral). However, its order clearly stays the same, i.e. O(1). A further error might arise in the inner sum. In every summation step we can miscount one integer which, together with the fact that we have kX summation steps (for a constant k), leads to an error of the order O(X).
The slightly harder to handle and, as we will see, more significant (but still negligible) source of errors is the transition from the inner sum to the integral, which can fortunately be investigated by applying the Euler-Maclaurin formula (also known as Euler's summation formula) which we will present with a proof from Apostol, 1998.
4.4, Fields with more than one related Minimal Pair (B,C):
With the above considerations and some more calculations we can now see more clearly that the number of minimal pairs (B,C) is exactly what we are interested in. This is because the number of fields with more than one related (minimal) pair (B,C) is of negligible order O(X2). We start with the definition of a primitive vector and, based on that, we bound the number of fields with the above property in terms of the discriminant.
5.2, The Class Number Formula and Other Convergences:
We will now introduce the class number formula which presents a connection between several invariant values of a number field, namely its class number, regulator, discriminant, and signature and the Dedekind zeta function. The formula is presented without
3.3, Estimation of Errors:
Let us first consider the easier-to-handle source of errors in the above formulas. The first error that comes up is because we use the floor and ceiling functions in the summation bounds in (3.1) which gives only an error of constant order O(1) for any possible choice of a. This error transfers to the expression via integral in (3.2) where we drop the ceiling function in the lower bound (we do that to avoid nasty results when computing the integral). However, its order clearly stays the same, i.e. O(1). A further error might arise in the inner sum. In every summation step we can miscount one integer which, together with the fact that we have kX summation steps (for a constant k), leads to an error of the order O(X).
The slightly harder to handle and, as we will see, more significant (but still negligible) source of errors is the transition from the inner sum to the integral, which can fortunately be investigated by applying the Euler-Maclaurin formula (also known as Euler's summation formula) which we will present with a proof from Apostol, 1998.
4.4, Fields with more than one related Minimal Pair (B,C):
With the above considerations and some more calculations we can now see more clearly that the number of minimal pairs (B,C) is exactly what we are interested in. This is because the number of fields with more than one related (minimal) pair (B,C) is of negligible order O(X2). We start with the definition of a primitive vector and, based on that, we bound the number of fields with the above property in terms of the discriminant.
5.2, The Class Number Formula and Other Convergences:
We will now introduce the class number formula which presents a connection between several invariant values of a number field, namely its class number, regulator, discriminant, and signature and the Dedekind zeta function. The formula is presented without
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Bibliographische Angaben
- Autor: Gero Brockschnieder
- 2015, Erstauflage, 88 Seiten, Masse: 15,5 x 22 cm, Kartoniert (TB), Englisch
- Verlag: Anchor Academic Publishing
- ISBN-10: 3954893894
- ISBN-13: 9783954893898
Sprache:
Englisch
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