Results 1  10
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15
Explicit bounds for primes in residue classes
 Math. Comp
, 1996
"... Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K su ..."
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Cited by 17 (1 self)
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Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K such that p = σ, satis
Class Field Theory in Characteristic p, its Origin and Development
 the Proceedings of the International Conference on Class Field Theory (Tokyo
, 2002
"... Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new imp ..."
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Cited by 14 (5 self)
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Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new impetus was given by the seminal thesis of E. Artin (1921, published in 1924). In this exposition I shall report on the development during the twenties and thirties of our century, with emphasis on the emergence of class field theory for function elds. The names of F.K.Schmidt, H. Hasse, E. Witt, C. Chevalley (among others) are closely connected with that development.
Some studies on arithmetical chaos in classical and quantum mechanics
 Internat. J. Modern Phys
, 1993
"... Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithm ..."
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Cited by 7 (0 self)
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Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithmetic structure. It is shown that the arithmetical features of the considered systems lead to exceptional properties of the corresponding spectra of lengths of closed geodesics (periodic orbits). The most significant one is an exponential growth of degeneracies in these geodesic length spectra. Furthermore, the arithmetical systems are distinguished by a structure that appears as a generalization of geometric symmetries. These pseudosymmetries occur in the quantization of the classical arithmetic systems as Hecke operators, which form an infinite algebra of selfadjoint operators commuting with the Hamiltonian. The statistical properties of quantum energies in the arithmetical systems have previously been identified as exceptional. They do not fit into the general scheme of random matrix theory. It is shown with the help of a simplified model for the spectral form factor
The BrauerHasseNoether theorem in historical perspective.
, 2001
"... this article in preparation for my talk at the conference March 2224, 2001 which is dedicated to the memory of Richard Brauer on the occasion of his 100. birthday ..."
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Cited by 5 (1 self)
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this article in preparation for my talk at the conference March 2224, 2001 which is dedicated to the memory of Richard Brauer on the occasion of his 100. birthday
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how ..."
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Cited by 5 (2 self)
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
Results and estimates on pseudopowers
 Math. Comp
, 1996
"... Abstract. Let n be a positive integer. We say n looks like a power of 2moduloaprime pif there exists an integer ep ≥ 0 such that n ≡ 2 ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a power ..."
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Cited by 3 (0 self)
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Abstract. Let n be a positive integer. We say n looks like a power of 2moduloaprime pif there exists an integer ep ≥ 0 such that n ≡ 2 ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a powerof2. Next, we define an xpseudopower of the base 2tobeapositiveintegern that is not a power of 2, but looks like a power of 2 modulo all primes p ≤ x. Let P2(x) denote the least such n. We give an unconditional upper bound on P2(x), a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that P2(x)isaboutexp(c2x/log x) for a certain constant c2. We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than 2 are also given. 1.
Numerical evidence for a conjectural generalization of Hilbert's theorem 132
 J. COMPUT. MATH.
, 2003
"... We develop an algorithm for computing numerical evidence for a conjecture whose validity is predicted by the requirement that the Equivariant Tamagawa Number conjectures for Tate motives as formulated by Burns and Flach are compatible with the functional equation of Artin Lseries. The algorithm inc ..."
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Cited by 3 (2 self)
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We develop an algorithm for computing numerical evidence for a conjecture whose validity is predicted by the requirement that the Equivariant Tamagawa Number conjectures for Tate motives as formulated by Burns and Flach are compatible with the functional equation of Artin Lseries. The algorithm includes methods for the computation of Fitting ideals and projective lattices over the integral group ring.
History of Valuation Theory  Part I
"... The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kursch'ak who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of padic fields as defined by Kurt Hensel. In the following decades we can o ..."
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Cited by 2 (0 self)
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The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kursch'ak who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of padic fields as defined by Kurt Hensel. In the following decades we can observe a quick development of valuation theory, triggered mainly by the discovery that much of algebraic number theory could be better understood by using valuation theoretic notions and methods. An outstanding figure in this development was Helmut Hasse. Independent of the application to number theory, there were essential contributions to valuation theory given by Alexander Ostrowski, published 1934. About the same time Wolfgang Krull gave a more general, universal definition of valuation which turned out to be applicable also in many other mathematical disciplines such as algebraic geometry or functional analysis, thus opening a new era of valuation theory.
Algebraic Number Theory
, 2009
"... 2. Number fields........................................ 9 3. Norms, traces and discriminants.............................. 15 4. Rings of integers....................................... 20 ..."
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2. Number fields........................................ 9 3. Norms, traces and discriminants.............................. 15 4. Rings of integers....................................... 20
6.3.1 Arithmetic of algebras and Hensel’s methods........ 44
"... 5.4.3 Algebras with pure maximal subfields............ 32 ..."