A differential criterion for complete intersections.
A Diophantine problem on algebraic curves over function fields of positive characteristic
A duality theorem in the multivariable Iwasawa theory of local fields.
A family of infinite pairs of quadratic fields ℚ(√D) and ℚ(√-D) whose class numbers are both divisible by 3
A finiteness theorem for imaginary abelian number fields.
A further note on the class number of real quadratic fields
A further omega result for the ellipsoid problem in algebraic number fields
A general framework for subexponential discrete logarithm algorithms
A generalisation of Artin's conjecture for primitive roots
A generalisation of Mahler measure and its application in algebraic dynamical systems
We prove a generalisation of the entropy formula for certain algebraic -actions given in [2] and [4]. This formula expresses the entropy as the logarithm of the Mahler measure of a Laurent polynomial in d variables with integral coefficients. We replace the rational integers by the integers in a number field and examine the entropy of the corresponding dynamical system.
A generalization of a problem of Chebyshev
A generalization of a result on integers in metacyclic extensions
Let p be an odd prime and let c be an integer such that c>1 and c divides p-1. Let G be a metacyclic group of order pc and let k be a field such that pc is prime to the characteristic of k. Assume that k contains a primitive pcth root of unity. We first characterize the normal extensions L/k with Galois group isomorphic to G when p and c satisfy a certain condition. Then we apply our characterization to the case in which k is an algebraic number field with ring of integers ℴ, and, assuming some...
A generalization of a theorem of Euler for regular chains of complex quadratic irrationalities
A generalization of a theorem of Minkowski
A generalization of a theorem of Schinzel
We give lower bounds for the Mahler measure of totally positive algebraic integers. These bounds depend on the degree and the discriminant. Our results improve earlier ones due to A. Schinzel. The proof uses an explicit auxiliary function in two variables.
A generalization of a third irreducibility theorem of I. Schur
A generalization of a unit index of Greither
A generalization of Bombieri's prime number theorem to algebraic number fields
A generalization of Davenport's constant and its arithmetical applications
A generalization of Dirichlet's unit theorem
We generalize Dirichlet's S-unit theorem from the usual group of S-units of a number field K to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over S. Specifically, we demonstrate that the group of algebraic S-units modulo torsion is a ℚ-vector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, and that the elements of this vector space which are linearly independent over ℚ retain their linear independence...