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28
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 16 (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 cases of the FontaineMazur conjecture
 J. Number Theory
, 1992
"... Abstract. We prove more special cases of the FontaineMazur conjecture regarding padic Galois representations unramified at p, and we present evidence for and consequences of a generalization of it. 0. Introduction. Answering a longstanding question of Furtwängler, mentioned as early as 1926 [10], ..."
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Cited by 12 (3 self)
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Abstract. We prove more special cases of the FontaineMazur conjecture regarding padic Galois representations unramified at p, and we present evidence for and consequences of a generalization of it. 0. Introduction. Answering a longstanding question of Furtwängler, mentioned as early as 1926 [10], Golod and Shafarevich showed in 1964 [8] that there exists a number field with an infinite, everywhere unramified prop extension. In fact it is easy to obtain many examples [8], [13], [23]. Very little is known, however, regarding the structure
The Galois theory of periodic points of polynomial maps
 Proc. London Math. Soc. 68
, 1994
"... There is a growing body of evidence that problems associated with iterated maps can yield interesting insights in number theory and algebra. There are the papers of Narkiewicz [1922], Lewis [14], Liardet [15], and more recently, Silverman [27], dealing with the diophantine aspects of iterated maps, ..."
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Cited by 11 (0 self)
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There is a growing body of evidence that problems associated with iterated maps can yield interesting insights in number theory and algebra. There are the papers of Narkiewicz [1922], Lewis [14], Liardet [15], and more recently, Silverman [27], dealing with the diophantine aspects of iterated maps, as well as the papers
conjectures and Hilbert’s twelfth problem. Experiment
 Math. 9
, 1996
"... We give a constructive proof of a theorem given in [Tate 84] which states that (under Stark’s Conjecture) the field generated over a totally real field K by the Stark units contains the maximal real Abelian extension of K. As a direct application of this proof, we show how one can compute explicitly ..."
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Cited by 11 (8 self)
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We give a constructive proof of a theorem given in [Tate 84] which states that (under Stark’s Conjecture) the field generated over a totally real field K by the Stark units contains the maximal real Abelian extension of K. As a direct application of this proof, we show how one can compute explicitly real Abelian extensions of K. We give two examples. In a series of important papers [Stark 71, Stark 75, Stark 76, Stark 80] H. M. Stark developed a body of conjectures relating the values of Artin Lfunctions at s = 1 (and hence, by the functional equation, their leading terms at s = 0) with certain algebraic quantities attached to extensions of number fields. For example, in the case of Abelian Lfunctions with a firstorder zero at s = 0, the conjectural relation is between the first derivative of the Lfunctions and the logarithmic embedding of certain units in ray class fields known as Stark units, which are predicted to exist. The use of these conjectures to provide explicit generators of ray class fields,
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
Some families of noncongruent numbers
"... Abstract. In this article we study the TateShafarevich groups corresponding to 2isogenies of the curve Ek: y 2 = x(x 2 − k 2) and construct infinitely many examples where these groups have odd 2rank. Our main result is that among the curves Ek, where k = pl ≡ 1 mod 8 for primes p and l, the curve ..."
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Cited by 5 (4 self)
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Abstract. In this article we study the TateShafarevich groups corresponding to 2isogenies of the curve Ek: y 2 = x(x 2 − k 2) and construct infinitely many examples where these groups have odd 2rank. Our main result is that among the curves Ek, where k = pl ≡ 1 mod 8 for primes p and l, the curves with rank 0 have density ≥ 1 2. 1.
Five peculiar theorems on simultaneous representation of primes by quadratic forms
 J. Number Theory
, 2009
"... It is a theorem of Kaplansky that a prime p ≡ 1 (mod 16) is representable by both or none of x 2 + 32y 2 and x 2 + 64y 2, whereas a prime p ≡ 9 (mod 16) is representable by exactly one of these binary quadratic forms. In this paper five similar theorems are proved. As an example, one theorem states ..."
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Cited by 2 (2 self)
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It is a theorem of Kaplansky that a prime p ≡ 1 (mod 16) is representable by both or none of x 2 + 32y 2 and x 2 + 64y 2, whereas a prime p ≡ 9 (mod 16) is representable by exactly one of these binary quadratic forms. In this paper five similar theorems are proved. As an example, one theorem states that a prime p ≡ 1 (mod 20) is representable by both or none of x 2 + 20y 2 and x 2 + 100y 2, whereas a prime p ≡ 9 (mod 20) is representable by exactly one of these forms. A heuristic argument is given why there are no other results of the same kind. The latter argument relies on the (highly plausible) conjecture that there are no negative discriminants ∆ other than the 485 known ones such that the class group C (∆) has exponent 4. The methods are purely classical. Consider a negative integer ∆ ≡ 0, 1 (mod 4) and recall that the principal form1 F (x, y) of discriminant ∆ is x2 − ∆ 4 y2 or x2 + xy − ∆−1 4 y2 according to ∆’s parity. It is well known that the prime numbers representable by F (x, y) are describable by congruence conditions if and only if each genus of forms of discriminant ∆ consists of a single class, or – equivalently – the class group C (∆) is either trivial or has exponent 2 (see e.g. [2, p. 62]). The determination of the negative discriminants with one class per genus is a famous problem in number theory. Gauss, who considered only forms of type ax2 + 2bxy + cy2, found 65 such discriminants −4n and showed that they correspond to Euler’s idoneal or convenient numbers n. Dickson [3] compiled a
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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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.
TABLES OF OCTIC FIELDS WITH A QUARTIC SUBFIELD
, 1999
"... We describe the computation of extended tables of degree 8 fields with a quartic subfield, using class field theory. In particular we find the minimum discriminants for all signatures and for all the possible Galois groups. We also discuss some phenomena and statistics discovered while making the t ..."
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We describe the computation of extended tables of degree 8 fields with a quartic subfield, using class field theory. In particular we find the minimum discriminants for all signatures and for all the possible Galois groups. We also discuss some phenomena and statistics discovered while making the tables, such as the occurrence of 11 nonisomorphic number fields having the same discriminant, or several pairs of nonisomorphic number fields having the same Dedekind zeta function.