Linear independence of harmonic numbers over the field of algebraic numbers II

The Ramanujan Journal - In a recent article, the authors have studied the arithmetic natures of a certain class of harmonic numbers. The present paper will explore the linear independence of...     

On algebraic independence of some numbers

A theorem is proved about algebraic independence of values of the exponential function.Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 525–532, November, 1968.     

On the transcendence and algebraic independence of certain continued fractions

The transcendence of continued fractions =[a ₀;a ₁,a ₂,...] is proved under growth conditions involving the denominatorsq _n of the convergents and shifted partial quotientsa _n+k . Extending this idea, conditions for the algebraic independence of several continued fractions are given. The proofs use the approximation properties of continued fractions in combination with the Thue-Siegel-Roth Theorem or a criterion for algebraic independence of Bundschuh.     

Concerning algebraic independence of some transcendental numbers

Given the three numbers of,a ^{1 2} , and Ina ₂/Ina ₁, wherea ₁ anda ₂ are algebraic numbers whose logarithms are linearly independent in a rational field and is a quadratic irrationality, it is shown that they are not all expressible algebraically in terms of one of them.Translated from Matematicheskie Zametki, Vol.3, No. 1, pp. 51–58, January, 1968.     

On the question of the algebraic independence of algebraic powers of algebraic numbers

We obtain results showing that transcendental numbers of the form , wherea0, 1, is irrational, anda and are algebraic numbers, cannot be expressed algebraically in terms of two of the numbers. The proof is carried out by A. O. Gel'fond's method.Translated from Matematicheskie Zametki, Vol. 11, No. 6, pp. 635–644, June, 1972.     

Center of the semigroup algebra of a finite inverse semigroup over the field of complex numbers

In the semigroup algebra A of a finite inverse semigroup S over the field of complex numbers to an indempotent e there is assigned the sum (e)=e+(–1)^Ke_L1e_iK, where e_i,...,e_m are maximal preidempotents of the idempotent e, and the summation goes over all nonempty subsets {i₁,...,i_k} of the set {1,...m} Then for any class K of conjugate group elements of the semigroup S the element K=a·(a^–1a) (the summation goes over all ag) is a central element of the algebra A, and the set {K} of all possible such elements is a basis for the center of the algebra A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 154–158, 1978.     

The Geisser-Levine method revisited and algebraic cycles over a finite field

We reformulate part of the arguments of T. Geisser and M. Levine relating motivic cohomology with finite coefficients to truncated étale cohomology with finite coefficients [9,10]. This reformulation amounts to a uniqueness theorem for motivic cohomology, and shows that the Geisser-Levine method can be applied generally to compare motivic cohomology with other types of cohomology theories. We apply this to prove an equivalence between conjectures of Tate and Beilinson on cycles in characteristic p and a vanishing conjecture for continuous étale cohomology. Received: 23 November 2000 / Published online: 5 September 2002     

Criteria of algebraic independence with multiplicities and interpolation determinants

We generalize Gel'fond's criterion of algebraic independence by taking into account the values of the derivatives of the polynomials, and show how the new criterion applies to proving results of algebraic independence using interpolation determinants. We also establish a new result of approximation of a transcendental number by algebraic numbers of bounded degree and size. It contains an earlier result of E. Wirsing and also a result announced by A. Durand.

     

Class numbers of algebraic number fields of Eisenstein type,II

Let K be an algebraic number field, of degree n, with a completely ramifying prime p, and let t be a common divisor of n and $\text{(p ? 1)}\text{2}$. Then it is proved that if K does not contain the unique subfield, of degree t, of the p-th cyclotomic number field, then we have (h_K, n) > 1, where h_K is the class number of K. As applications, we give several results on h_K of such algebraic number fields K.     

Factorization of polynomials over a finite field and the solution of systems of algebraic equations

An algorithm is constructed for factoring polynomials in several variables over a finite field, which works in polynomial time in the size of the polynomial and q. Previously this result was known in the case of one variable. An algorithm is given for the solution (over the algebraic closure F of the field F) of systems of algebraic equations, where with working time of order where L is the size of a representative of the original system, is the degree of transcendence of the field F over the prime subfield, q=char(F). Previously the estimate was known for =0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 137, pp. 20–79, 1984.     

The algebraic independence of certain numbers related by the exponential function

We give a variation of a theorem of Gelfond. One of the corollaries is Schneider's conjecture that either exp(e) or exp(e²) is transcendental. It also follows that at least one of the numbers exp(α), exp(α²), exp(α³) is transcendental for any nonzero complex number α.