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 degree-1 prime p of K such that p = σ, satis-

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.

Abstract. We prove more special cases of the Fontaine-Mazur conjecture regarding p-adic 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 pro-p extension. In fact it is easy to obtain many examples [8], [13], [23]. Very little is known, however, regarding the structure

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 [19-22], Lewis [14], Liardet [15], and more recently, Silverman [27], dealing with the diophantine aspects of iterated maps, as well as the papers

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 L-functions 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 L-functions with a first-order zero at s = 0; the conjectural relation is between the first derivative of the L-functions 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 generat...

this article in preparation for my talk at the conference March 22-24, 2001 which is dedicated to the memory of Richard Brauer on the occasion of his 100. birthday

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

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 p-adic 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.

Abstract. 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 non-isomorphic number fields having the same discriminant, or several pairs of non-isomorphic number fields having the same Dedekind zeta function. 1.

I begin with a short synopsis of the classfield-theoretic facts about Abelian number fields essential for all that follows, which are largely justified in detail in my Klassenkörperbericht or which result from the general class field theorems by specializing to the Abelian case. I restrict myself here to the main features. I state a few facts going into more detail in the sequel when the occasion arises. The notation that I introduce in this synopsis, as well as a few further terms introduced in the following derivation of the general class number formula, will then be continued to be used without explaining it each time. Let K be an Abelian number field of degree n over the rational numbers Q. It is class field to a rational congruence subgroup H of index n. Let f be the conductor of H, also called the conductor of K. Then H consists of all numbers in a group of prime residue classes mod f, the n classes for H consisting of the numbers of its n cosets in the group of all prime residue classes mod f, and this division into classes can not be described using the prime residue classes of a proper divisor of f. The class field property of K says that, in the manner determined by the class