Simultaneous rational approximations to certain algebraic numbers

1967 ◽  
Vol 63 (3) ◽  
pp. 693-702 ◽  
Author(s):  
A. Baker
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Infinitely Many Solutions ◽  
Rational Approximations ◽  
Algebraic Numbers ◽  
Famous Theorem ◽  
Image Position

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the formis satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such thatThis result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

scholarly journals On the approximation of algebraic numbers by algebraic integers

1963 ◽  
Vol 3 (4) ◽  
pp. 408-434 ◽  
Author(s):  
K. Mahler
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Positive Constant ◽  
Algebraic Number Field ◽  
Algebraic Numbers ◽  
Algebraic Integers ◽  
Image Position ◽  
Rational Integral

In his Topics in Number Theory, vol. 2, chapter 2 (Reading, Mass., 1956) W. J. LeVeque proved an important generalisation of Roth's theorem (K. F. Roth, Mathematika 2,1955, 1—20).Let ξ be a fixed algebraic number, σ a positive constant, and K an algebraic number field of degree n. For κ∈K denote by κ(1), …, κ(n) the conjugates of κ relative to K, by h(κ) the smallest positive integer such that the polynomial has rational integral coefficients, and by q(κ) the quantity

Computing the effectively computable bound in Baker's inequality for linear forms in logarithms

1976 ◽  
Vol 15 (1) ◽  
pp. 33-57 ◽  
Author(s):  
A.J. van der Poorten ◽  
J.H. Loxton
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
Computable Bound ◽  
Image Position ◽  
Detailed Computation

For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation)have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.

A Periodic Jacobi-Perron Algorithm

1965 ◽  
Vol 17 ◽  
pp. 933-945
Author(s):  
Leon Bernstein
Keyword(s):  
Algebraic Equation ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Image Position

In the first part of this paper I shall demonstrate that one irrational root of the algebraic equationcreates an algebraic number field, out of which n — 1 irrationals can be chosen so that they yield a periodic Jacobi-Perron algorithm. The coefficients in (1) are subject to certain restrictions which will be elaborated below.

Uniform bounds for the number of solutions to Yn = f(X)

1986 ◽  
Vol 100 (2) ◽  
pp. 237-248 ◽  
Author(s):  
J.-H. Evertse ◽  
J. H. Silverman
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Solutions To Equations ◽  
Number Of Solutions ◽  
Explicit Result ◽  
Uniform Bounds ◽  
The Past ◽  
Image Position ◽  
Integral Solutions

Let K be an algebraic number field and f(X) ∈ K[X]. The Diophantine problem of describing the solutions to equations of the formhas attracted considerable interest over the past 60 years. Siegel [12], [13] was the first to show that under suitable non-degeneracy conditions, the equation (+) has only finitely many integral solutions in K. LeVeque[7] proved the following, more explicit, result. Letwhere a ∈ K* and αl,…,αk are distinct and algebraic over K. Then (+) has only finitely many integral solutions unless (nl,…,nk) is a permutation of one of the n-tuples

How to solve a quadratic equation in integers

1981 ◽  
Vol 89 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Fritz J. Grunewald ◽  
Daniel Segal
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Quadratic Equation ◽  
Effective Procedure ◽  
Image Position

In answer to a question posed by J. L Britton in (2), we sketch in this note an effective procedure to decide whether an arbitrary quadratic equationwith rational coefficients, has a solution in integers. A similar procedure will in fact decide whether such an equation over a (suitably specified) algebraic number field k has a solution in any (suitably specified) order in k; but we shall not burden the exposition by giving chapter and verse for this claim.

scholarly journals The Brauer group of cubic surfaces

1993 ◽  
Vol 113 (3) ◽  
pp. 449-460 ◽  
Author(s):  
Sir Peter Swinnerton-Dyer
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Rational Surface ◽  
Hasse Principle ◽  
Weak Approximation ◽  
Brauer Group ◽  
Cubic Surfaces ◽  
Image Position

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injectionis an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

scholarly journals Groups of Matrices With Integer Eigenvalues

1971 ◽  
Vol 12 (3) ◽  
pp. 351-357 ◽  
Author(s):  
M. R. Freislich
Keyword(s):  
Number Field ◽  
Characteristic Polynomial ◽  
Linear Group ◽  
Algebraic Number ◽  
General Linear Group ◽  
Algebraic Number Field ◽  
General Linear ◽  
Algebraic Numbers ◽  
Algebraic Integers

Let F be an algebraic number field, and S a subgroup of the general linear group GL(n, F). We shall call S a U-group if S satisfies the condition (U): Every x ∈ S is a matrix all of whose eigenvalues are algebraic integers. (This is equivalent to either of the following conditions: a) the eigenvalues of each matrix (x are all units as algebraic numbers; b) the characteristic polynomial for x has all its coefficients integers in F. In particular, then, every group of matrices with entries in the integers of F is a U-group.

An Independent System of Units in Certain Algebraic Number Fields

1985 ◽  
Vol 37 (4) ◽  
pp. 644-663
Author(s):  
Claude Levesque
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Independent System ◽  
System Of Units ◽  
Image Position

For Kn = Q(ω) a real algebraic number field of degree n over Q such thatwith D ∊ N, d ∊ Z, d|D2, and D2 + 4d > 0, we proved in [5] (by using the approach of Halter-Koch and Stender [6]) that ifwiththenis an independent system of units of Kn.

Waring's problem in algebraic number fields

1961 ◽  
Vol 57 (3) ◽  
pp. 449-459 ◽  
Author(s):  
B. J. Birch
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Good Deal ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Singular Series ◽  
Algebraic Number Fields ◽  
Extra Information ◽  
Totally Positive

Let K be a finite algebraic number field, of degree R. Then those integers of K which may be expressed as a sum of dth powers generate a subring JK, d of the integers of K (JK, d need not be an ideal of K, as the simplest example K = Q(i), d = 2 shows. JK, d is an order, it in fact contains all integer multiples of d!; it also contains all rational integers). Siegel(12) showed that every sufficiently large totally positive integer of JK, d is the sum of at most (2d−1 + R) Rd totally positive dth powers; and he conjectured that the number of dth powers necessary should be independent of the field K—for instance, he had proved(11) that five squares are enough for every K. In this paper, we will show that, as far as the analytic part of the argument is concerned, Siegel's conjecture is correct. I have not been able to deal properly with the problem of proving that the singular series is positive; but since Siegel wrote, a good deal of extra information about singular series has been obtained, in particular by Stemmler(14) and Gray(4). The most spectacular consequence of all this is that if p is prime, then every large enough totally positive integer of JK, p is a sum of (2p + 1) totally positive pth. powers.

scholarly journals A note on a result of Mahler's

1966 ◽  
Vol 6 (4) ◽  
pp. 399-401
Author(s):  
Indar S. Luthar
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Image Position

In a recent paper, Maher [2] proved that for any algebraic number field K of degree n and discriminant d there exists a constant C depending only on n and d such that for any ceiling λP of K there exists a basis α1 …, αn of the corresponding ideal αλ such that .

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