Real Algebraic Number Research Articles

Regulatorabschätzungen für total reelle algebraische Zahlkörper

A lower bound for the regulator of a totally real algebraic number field is determined. The regulator occurs in the determinant of a suitable positive definite quadratic form, and the desired bound is obtained by estimating the minimum of this quadratic form from below. This can be done by solving an extremal value problem with subsidiary conditions from the properties of the units of the field. Numerous examples (Tables I, III) illustrate the advantage of this method over known results.

Scalar extension of quadratic lattices II

Let k be a totally real algebraic number field, the maximal order of k, and let L (resp. M) be a Z-lattice of a positive definite quadratic space U (resp. V) over the field Q of rational numbers. Suppose that there is an isometry σ from L onto M. We have shown that the assumption implies σ(L) = M in some cases in [2]. Our aim in this paper is to improve the results of [2]. In § 1 we introduce the notion of E-type: Let L be a positive definite quadratic lattice over Z.

Linearformen mit algebraischen Koeffizienten

In a recent paper I generalized the Thue-Siegel-Roth-Schmidt theorem on simultaneous rational approximation of n real algebraic numbers to include also the p-adic case. In the present paper I shall carry the argument further and prove analogous results for systems of real and p-adic linear forms with algebraic coefficients.

The Fourier expansion of Epstein's zeta function for totally real algebraic number fields and some consequences for Dedekind's zeta function

The Fourier expansion of Epstein's zeta function for totally real algebraic number fields and some consequences for Dedekind's zeta function

Units and periodic Jacobi-Perron algorithms in real algebraic number fields of degree $3$

It is not known whether or not the Jacobi-Perron Algorithm of a vector in ${R_{n - 1}},n \geqslant 3$, whose components are algebraic irrationals, always becomes periodic. The author enumerates, from his previous papers, a few infinite classes of real algebraic number fields of any degree for which this is the case. Periodic Jacobi-Perron Algorithms are important, because they can be applied,

On the Representation in Sums of Squares for Definite Functions in One Variable Over an Algebraic Number Field

1. If R is a real closed field, then Artin [1] showed that every positive definite function f(Xl, . . ., Xj) in the rational function field R (X1,.. ., X.,) may be expressed as a sum of squares, thus solving a problem of Hilbert. However, Artin did not furnish the number of necessary squares. It was Pfister [8] who proved that 2' number of squares would suffice. Whether or not this upper bound is the best possible in general is still an outstanding open question, although for n < 2 this has been settled in the affirmative by Cassels-EllisonPfister [3]. To generalize this Hilbert problem with another ground field K instead of the real closed field R, Pfister asked in [8] whether 2n+2 might serve as an upper bound for K = Q, the rational number field. For n = 0, the theorem of Euler-Lagrange provides the answer. For n =1, Landau [5] showed that every positive definite polynomial in one variable over Q is a sum of eight squares of polynomials. Pourchet, however, recently proved [9] the rather startling fact that five is, in fact, the best possible bound in this case. In this short note we complete Pourchet's work by explicitly determining the best possible bound for K (X) where K is any formally real algebraic number field. We shall call this invariant the height. Thus, the reduced height m(F) of a field F is the least positive integer (or infinity) such that every sum of squares in F is a sum of m(F) number of squares. More precisely, we prove the following:

L-series, modular imbeddings, and signatures

For each cusp singularity of the Hilbert modular variety of a totally real algebraic number field there is associated an L-series which determines the corresponding to the version of Selberg's trace formula worked out by Shimizu. For a real quadratic field Meyer [9, 10] has given an elementary algorithm by Dedekind sums for the computation of the parabolic contribution (see also Siegel [14]) from which a simple formula for the parabolic contribution in terms of the continued fraction associated with the cyclic resolution of the corresponding cusp singularity can be derived ([5, 7, 8]). In fact, the parabolic contribution equals ~2(3r-bo b l . . . . . br-1) where bo .. . . . br-1 are the characteristic numbers of the dual normal bundles of the rational curves which form the exceptional fibre of the cyclic resolution. The first purpose of this note is to give a simple formula for the total parabolic contribution, i.e., the sum of the parabolic contributions for the different cusps. This formula is deduced from a general identity satisfied by certain L-series in any real algebraic number field. In the quadratic case one obtains an expression for the total parabolic contribution in terms of the class numbers of imaginary quadratic fields. Further, in the quadratic case the total parabolic contribution vanishes if and only if the discriminant of the field is the sum of two squares and is negative otherwise. The total parabolic contribution is of additional interest in the quadratic case for two reasons: first, it is one-fourth the signature (adjusted for defects coming from quotient singularities in certain special cases) of the open rational homology 4-manifold obtained as the quotient of the product of two upper-half planes by the non-symmetric Hilbert modular group. This result, announced to some extent in [5], will be proved here. Thus, from [2] the vanishing of this (adjusted) signature is a necessary and sufficient condition for the existence of modular imbeddings. Second, it will be shown that this signature is twice the difference of the arithmetic genera for the ordinary and mixed Hilbert modular groups.

On a conjecture of Hecke concerning elementary class number formulas

Let K be a totally real algebraic number field of class number hK and L a totally imaginary quadratic extension of K of class number hL. Hecke conjectured that there exists an “elementary” formula for the first factor hL/hK of the class number of L. The paper develops a theory which allows computation of hL/hK in terms of the periods of certain complex differential forms associated to a manifold defined in a natural way from K. Thus, Hecke's conjecture is reduced to the problem of finding elementary formulas for these periods. The essential idea of the proof consists of establishing a Kronecker limit formula for the non-analytic Eisenstein series for the Hilbert modular group for K.

Diophantine approximation of linear forms over an algebraic number field

This paper gives an algorithm for generating all the solutions in integers x0, x1…, xn of the inequalitywhere 1, α1, …,αn are numbers, linearly independent over the rationals, in a real algebraic number field of degree n + 1 ≥ 3 and c is any sufficiently large positive constant. It is well known [2, p. 79] that if c is small enough, then (1) has no integer solutions with x1…, xn not all zero.

Class Numbers of Totally Imaginary Quadratic Extensions of Totally Real Fields

Let K be a totally real algebraic number field. This paper provides an effective constant $C(K,h)$ such that every totally imaginary quadratic extension L of K with ${h_L} = h$ satisfies $|{d_L}| < C(K,h)$ with at most one possible exception. This bound is obtained through the determination of a lower bound for $L(1,\chi )$ where $\chi$ is the ideal character of K associated to L. Results of Rademacher concerning estimation of L-functions near $s = 1$ are used to determine this lower bound. The techniques of Tatuzawa are used in the proof of the main theorem.

Class numbers of totally imaginary quadratic extensions of totally real fields

Let K be a totally real algebraic number field. This paper provides an effective constant $C(K,h)$ such that every totally imaginary quadratic extension L of K with ${h_L} = h$ satisfies $|{d_L}| < C(K,h)$ with at most one possible exception. This bound is obtained through the determination of a lower bound for $L(1,\chi )$ where $\chi$ is the ideal character of K associated to L. Results of Rademacher concerning estimation of L-functions near $s = 1$ are used to determine this lower bound. The techniques of Tatuzawa are used in the proof of the main theorem.

A Continued Fraction Algorithm for Real Algebraic Numbers

A Continued Fraction Algorithm for Real Algebraic Numbers

A continued fraction algorithm for real algebraic numbers

Let a denote a real algebraic number that is a root of a polynomial f ( x ) ∈ Z [ x ] f(x) \in {\text {Z}}[x] . The purpose of this paper is to state an algorithm for finding the simple continued fraction expansion of α \alpha . Furthermore, an application of the algorithm to sign determination in real algebraic number fields is given.

The Genesis and Development of Set Theory

U, If one year can be specified as the time when set theory started, that year should probably be 1874, the year in which Georg Cantor's paper [2] was published establishing the countability of the set of real algebraic numbers and the noncountability of the set of real numbers. The proof that the set of real numbers is not countable used nested intervals instead of Cantor's famous diagonal process, which appeared in a later work [9]. According to Fraenkel, Cantor himself had first thought that the continuum could be put in one-to-one correspondence with the set of natural numbers [12, p. 237]. Cantor observed in the introduction of his paper that combining the two theorems gives a result first proved by Liouville-that in each given interval there exist infinitely many transcendental (nonalgebraic) real numbers. There was some difficulty surrounding the publication of Cantor's next article [3] in 1878. The work remained in the publishing room of Crelle's Journal longer than usual for that time, apparently because Cantor's ideas were rejected by Leopold Kronecker, who was on the editing staff of the journal [12, p. 198]. The work was eventually published in the journal, but it was the last article Cantor published there.

Simultaneous Asymptotic Diophantine Approximations to a Basis of a Real Number Field

The purpose of this paper is to prove the following result.Theorem 1. Let K be a real algebraic number field of degree m = n + 1. Let 1, β1, …, βn be a basis of K.

A Note on the Brauer-Speiser Theorem

The Brauer-Speiser theorem asserts that the Schur index of a real-valued complex irreducible character of a finite group is either $1$ or $2$. In this paper we present a brief proof of this result. From this it follows that the $K$-central nontrivial division algebra components of group algebras over a real algebraic number field $K$ are quaternions.

A note on the Brauer-Speiser theorem

The Brauer-Speiser theorem asserts that the Schur index of a real-valued complex irreducible character of a finite group is either $1$ or $2$. In this paper we present a brief proof of this result. From this it follows that the $K$-central nontrivial division algebra components of group algebras over a real algebraic number field $K$ are quaternions.

Computable fields and arithmetically definable ordered fields

Introduction. A computable field is one whose elements may be placed in one-one correspondence with the natural numbers in such a way that the number theoretic functions corresponding to the field operations are recursive. In the same vein a field is called arithmetically definable (AD for short) if its elements may be placed in one-one correspondence with the natural numbers in such a way that the number theoretic functions corresponding to the field operations are arithmetical. These notions clearly extend in an obvious way to ordered fields and indeed to algebraic structures in general. The term computable structure (group, ring, etc.) was probably introduced for the first time by M. 0. Rabin [4], however, a similar notion was discussed a few years earlier by Frohlich and Shepherdson [1]. Each of these references contains a number of interesting theorems on computable structures. Some results concerning AD structures appear in [2 ]. The main purpose of the present paper is to show that the fields of real algebraic numbers, constructible numbers, and solvable numbers, which were shown to be AD in [2], are in fact computable. This answers a question raised in footnote (2) of [2 ].

Über die werte der ringklassen-L-funktionen reell-quadratischer zahlkörper an natürlichen argumentstellen

As E. Hecke (1924) conjectured and H. Klingen (1962) has proved, a certain class of L-series with class field theoretical characters associated with totally real algebraic number fields takes values (for natural argument) which are rational numbers save for well-known irrational factors. Klingen's method for proof, however, does not provide explicit formulas of elementary nature for those values. In this paper for the case of ring classes in real quadratic number fields formulas of that kind are obtained by starting with Hecke's integral representation of class-zeta-functions.

Local Representations of Galois Groups

The purpose of this paper is to present some infinite Galois extensions of algebraic number fields, whose Galois groups possess some matrix representations in local fields, through which the Frobenius automorphisms have algebraic numbers of absolute value N(p)A/2 as characteristic roots. Here p is the prime ideal of the basic field, for which the Frobenius automorphism is considered, and Ix is an integer determined by the representation. These Galois extensions will be generated by the coordinates of the points lying on some Galois coverings of an algebraic variety, which is isomorphic to the quotient of the Siegel upper half space t, by an arithmetic discontinuous group. In the last section of the previous paper [7], we have discussed certain local representations of the Galois groups of such extensions. The present results will give a re-formulation and a refinement of those obtained there.' To be more specific, we consider a certain reductive algebraic group 6 defined over Q which has the following properties: ( 1 ) The center of oQ is isomorphic to the multiplicative group F X of a totally real algebraic number field F. ( 2 ) 63Q acts faithfully on a vector space X over F, where the action of Fx is the same as the center of ok through the isomorphism of (1). ( 3 ) The semi-simple part of 6R is isomorphic to the product of Sp (X1, R) and a compact group. By (3), we can find a subgroup OV of mQ of index 2, which acts on . Let xF denote the ring of algebraic integers in F, and X a finitely generated TFsubmodule of X which spans X over F. Define subgroups F and Fa of W, for every integral ideal a in F, by