Units In Algebraic Number Fields Research Articles

Dual Diophantine Systems of Linear Inequalities

The paper suggests a modified version of the ℒ-algorithm for constructing an infinite sequence of integral solutions of dual systems $$ \mathcal{S} $$ and $$ {\mathcal{S}}^{\ast } $$ of linear inequalities in d + 1 variables consisting of k⊥ and k*⊥ inequalities, respectively, where k⊥ + k*⊥ = d + 1. Solutions are obtained from two recurrence relations of order d + 1. Approximation in the inequality systems $$ \mathcal{S} $$ and $$ {\mathcal{S}}^{\ast } $$ is effected with the Diophantine exponents $$ \frac{d+1-{k}^{\perp }}{k^{\perp }}-\upvarrho $$ and $$ \frac{d+1-{k}^{\ast \perp }}{k^{\ast \perp }}-\upvarrho $$ , respectively, where the deviation ϱ > 0 can be made arbitrarily small by appropriately choosing the recurrence relations. The ℒ-algorithm is based on a method for localizing units in algebraic number fields.

Dependency of units in number fields

We develop a method for validating the indepencence of units in algebraic number fields. In case that a given system of units has a dependency, we compute a certificate for this.

Distribution of units of algebraic number fields with only one fundamental unit

For some algebraic number fields $F$ with only one fundamental unit, we give a lower bound of the extension degree of the ray class field of conductor a rational prime $p$ over the Hilbert class field of $F$.

On the Units of Algebraic Number Fields

Let K be an algebraic number field and k be a proper subfield of K. Then we have the relations between the relative degree [K : k] and the increase of the rank of the unit groups. Especially, in the case of mth cyclotomic field Q(ζm), we determine the number m such that the increase of the rank of the unit groups is equal to the number of the imaginary conjugates of the proper subfield.

A Practical Version of the Generalized Lagrange Algorithm

We describe an implementation of the generalized Lagrange algorithm for computing units in algebraic number fields [Buchmann 1987a], together with extensive experimental data of the algorithm's application (to all totally real quartic fields of discriminant below 60000). We also present an improved algorithm, with related experimental data.

Parametrized units in algebraic number fields

One of the fundamental problems in algebraic number theory is the construction of units in algebraic number fields. Various authors have considered number fields which are parametrized by an integer variable. They have described units in these fields by polynomial expressions in the variable e.g. the fields ℚ(√[N2 + 1]), Nεℤ, with the units εN = N + √[N2 + l]. We begin this article by formulating a general principle for obtaining units in algebraic function fields and candidates for units in parametrized families of algebraic number fields. We show that many of the cases considered previously in the literature by such authors as Bernstein [2], Neubrand [8], and Stender [ll] fall in under this principle. Often the results may be obtained much more easily than before. We then examine the connection between parametrized cubic fields and elliptic curves. In §4 we consider parametrized quadratic fields, a situation previously studied by Neubrand [8]. We conclude in §5 by examining the effect of parametrizing the torsion structure on an elliptic curve at the same time.

A Remark on a Theorem of W. E. H. Berwick

A Remark on a Theorem of W. E. H. Berwick

On the calculation of the units of algebraic number fields

The method to find a system of fundamental units were given by G. F. Voronoi in the case of purely cubic fields (see B. N. Delone and D. K. Faddeev [1]), and by K. K. Biilebic [2] in general case of algebraic number fields except for quadratic fields. But they are rather complicated for direct calculation. On the other hand, in some special cases, units can be calculated by the Jacobi-Perron algorithm which is a generalization of the continued fractional expansion (see, for example, L. Bernstein [3]). In this paper, we will show the new method to calculate a system of independent units by multiplying several numbers. We will give some examples for purely cubic fields in Section 3.

Some new combinatorial identities derived from units in algebraic number fields

Two new combinatorial identities are derived from explicitly stated units in algebraic number fields of degree n = 3. Let e = w − D be a unit in a cubic field where w 3 = D 3 + 1, D ϵ N and n = 0,1, …. Then z n = t n+1 2 − t n t n +2 and t n = z n −1 2 − z n −2 z n are two combinatorial identities where t n+2 and z n+1 are obtained from some recursion formulas. Both have the same structure.

Diophantine equations and identities

The general diophantine equations of the second and third degree are far from being totally solved. The equations considered in this paper are i) x2−my2=±1 ii) x3+my3+m2z3−3mxyz=1iii) Some fifth degree diopantine equationsInfinitely many solutions of each of these equations will be stated explicitly, using the results from the ACF discussed before.It is known that the solutions of Pell's equation are well exploited. We include it here because we shall use a common method to solve these three above mentioned equations and the method becomes very simple in Pell's equations case.Some new third and fifth degree combinatorial identities are derived from units in algebraic number fields.

On effective computation of fundamental units. II

The new method for efficient computation of fundamental units of an algebraic number field originated by two of die authors in part I is used to develop a powerful computer program. Besides the description of this program part II contains a complete list of fundamental units of algebraic number fields of degree n = 3 , 4 , 5 , 6 n = 3,4,5,6 and small absolute discriminant.

Calculation of the regulator of a pure cubic field

A description is given of a modified version of Voronoi’s algorithm for obtaining the regulator of a pure cubic field Q ( D 3 ) Q(\sqrt [3]{D}) . This new algorithm has the advantage of executing relatively rapidly for large values of D. It also eliminates a computational problem which occurs in almost all algorithms for finding units in algebraic number fields. This is the problem of performing calculations involving algebraic irrationals by using only approximations of these numbers. The algorithm was implemented on a computer and run on all values of D ( ⩽ 10 5 ) D\;( \leqslant {10^5}) such that the class number of Q ( D 3 ) Q\;(\sqrt [3]{D}) is not divisible by 3. Several tables summarizing the results of this computation are also presented.

On some truncated units in algebraic number fields of degreen≧3

In this paper we present a partial answer to the following question. What algebraic number fields have truncated units others than of the form 1+xw+yw2.

Vektoren minimaler L�nge in Gittern mit einem Automorphismus

In a class of latticesM which are invariant under an orthogonal automorphism σ vectorsx≠0 of minimal Euclidean norm are considered. To obtain the finitely many possible indices (M:ℤ[〈σ〉]x)H. W. Leopoldt's norm inequality [5] and his constantc* are used. An effective procedure for the calculation ofc* is described; some values ofc* are listed. — An application to the theory of units of algebraic number fields is mentioned.

On an Algorithm of Billevich for Finding Units in Algebraic Number Fields

On an Algorithm of Billevich for Finding Units in Algebraic Number Fields

On an algorithm of Billevich for finding units in algebraic fields

The well-known algorithm of Billevich for finding units in algebraic number fields is derived by algebraic methods. Some tables of units in cubic and quartic fields are given.

Units of algebraic numberfields

Let K be an algebraic number field with proper subfield k. If K and k have the same number of fundamental units then relations between the units of K and k are obtained.

On the units of algebraic number fields

MathematikaVolume 14, Issue 2 p. 121-124 Research Article On the units of algebraic number fields Armand Brumer, Armand Brumer Department of Mathematics, Columbia University, New York, U.S.A.Search for more papers by this author Armand Brumer, Armand Brumer Department of Mathematics, Columbia University, New York, U.S.A.Search for more papers by this author First published: 01 December 1967 https://doi.org/10.1112/S0025579300003703Citations: 107AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume14, Issue2December 1967Pages 121-124 RelatedInformation

Note on characters of the groups of units of algebraic number fields

Note on characters of the groups of units of algebraic number fields

A Note on Units of Algebraic Number Fields

We shall prove in the present note a theorem on units of algebraic number fields, applying one of the strongest formulations, be Hasse [3], of Grunwald’s existence theorem.