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Let O be the ring of integers of a number field, and let n ≥ 3 . This paper studies bi-interpretability of the ring of integers Z with the special linear group SL n ( O ) , the general linear group GL n ( O ) and the subgroup T n ( O ) of GL n ( O ) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.

In this paper, we provide number-theoretic formulas for Farrell–Tate cohomology for SL 2 over rings of S-integers in number fields satisfying a weak regularity assumption. These formulas describe group cohomology above the virtual cohomological dimension, and can be used to study some questions in homology of linear groups. We expose three applications, to (I) detection questions for the Quillen conjecture, (II) the existence of transfers for the Friedlander–Milnor conjecture, (III) cohomology of SL 2 over number fields.

In their proof of the IP Szemeredi theorem, a far reaching extension of the classic theorem of Szemeredi on arithmetic progressions, Furstenberg and Katznelson [14] introduced an important class of additively large sets called IP r ⁎ sets which underlies recurrence aspects in dynamics and is instrumental to enhanced formulations of combinatorial results. The authors recently showed that additive IP r ⁎ subsets of Z d are multiplicatively rich with respect to every multiplication on Z d without zero divisors (e.g. multiplications induced by degree d number fields). In this paper, we explain the relationships between classes of multiplicative largeness with respect to different multiplications on Z d . We show, for example, that in contrast to the case for Z , there are infinitely many different notions of multiplicative piecewise syndeticity for subsets of Z d when d ≥ 2 . This is accomplished by using the associated algebra representations to prove the existence of sets which are large with respect to some multiplications while small with respect to others. In the process, we give necessary and sufficient conditions for a linear transformation to preserve a class of multiplicatively large sets. One consequence of our results is that additive IP r ⁎ sets are multiplicatively rich in infinitely many genuinely different ways. We conclude by cataloging a number of sources of additive IP r ⁎ sets from combinatorics and dynamics.

Given a number field k, we show that, for many finite groups G, all the Galois extensions of k with Galois group G cannot be obtained by specializing any given finitely many Galois extensions E / k ( T ) with Galois group G and E / k regular. Our examples include abelian groups, dihedral groups, symmetric groups, general linear groups over finite fields, etc. We also provide a similar conclusion while specializing any given infinitely many Galois extensions E / k ( T ) with Galois group G and E / k regular of a certain type, under a conjectural “uniform Faltings' theorem”.

After introducing the Ogus realization of 1-motives we prove that it is a fully faithful functor. More precisely, following a framework introduced by Ogus, considering an enriched structure on the de Rham realization of 1-motives over a number field, we show that it yields a full functor by making use of an algebraicity theorem of Bost.

Let G be a finite group and let N / E be a tamely ramified G -Galois extension of number fields whose inverse different C N / E is a square. Let A N / E denote the square root of C N / E . Then A N / E is a locally free Z [ G ] -module, which is in fact free provided N / E has odd order, as shown by Erez. Using M. Taylor's theorem, we can rephrase this result by saying that, when N / E has odd degree, the classes of A N / E and O N (the ring of integers of N ) in Cl ( Z [ G ] ) are equal (and in fact both trivial). We show that the above equality of classes still holds when N / E has even order, assuming that N / E is locally abelian. This result is obtained through the study of the Frohlich representatives of the classes of some torsion modules, which are independently introduced in the setting of cyclotomic number fields. Jacobi sums, together with the Hasse–Davenport formula, are involved in this study. Finally, when G is the binary tetrahedral group, we use our result in conjunction with Taylor's theorem to exhibit a tame G -Galois extension whose square root of the inverse different has nontrivial class in Cl ( Z [ G ] ) .

Given an integral domain A we consider the set of all integral elements over A that can occur as an eigenvalue of a symmetric matrix over A. We give a sufficient criterion for being such an element. In the case where A is the ring of integers of an algebraic number field this sufficient criterion is also necessary and we show that the size of matrices grows linearly in the degree of the element. The latter result settles a questions raised by Bass, Estes and Guralnick., Serious error in previous version

The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve E_f over K. In our previous paper, we associated, building on works of Darmon and Greenberg, a p-adic lattice to f, under certain hypothesis, and implicitly conjectured that this lattice is commensurable with the p-adic Tate lattice of E_f . In this paper, we present this conjecture in detail and discuss how it can be used to compute, directly from f, an explicit Weierstrass equation for the conjectural E_f . We develop algorithms to this end and implement them in order to carry out extensive systematic computations in which we compute Weierstrass equations of hundreds of elliptic curves, some with huge heights, over dozens of number fields. The data we obtain provide overwhelming amount of support for the conjecture and furthermore demonstrate that the conjecture provides an efficient tool to building databases of elliptic curves over number fields., Comment: Fixed citations. 32 pages, comments welcome

The aim of this paper is to investigate rotated versions of the densest known lattices in dimensions 2, 3, 4, 5, 6, 7, 8, 12 and 24 constructed via ideals and free Z -modules that are not ideals in subfields of cyclotomic fields. The focus is on totally real number fields and the associated full diversity lattices which may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. We also discuss on the existence of a number field K such that it is possible to obtain the lattices A 2 , E 6 and E 7 via a twisted embedding applied to a fractional ideal of O K .

Let R be the ring of algebraic integers in a number field K and let Λ be a maximal order in a finite dimensional semisimple K-algebra B. Building on our previous work [3], we compute the smallest number of algebra generators of Λ considered as an R-algebra. This reproves and vastly extends the results of P.A.B. Pleasants, who considered the case when B is a number field. In order to achieve our goal, we obtain several results about counting generators of algebras which have finitely many elements. These results should be of independent interest.

In this paper, we define Euclidean minima for function fields and give some bound for this invariant. We furthermore show that the results are analogous to those obtained in the number field case. (C) 2013 The Authors. Published by Elsevier Inc. All rights reserved.

We calculate the representation zeta function of the Heisenberg group over the integers of a quadratic number field. In general, the representation zeta function of a finitely generated torsion-free nilpotent group enumerates equivalence classes of representations, called twist-isoclasses. This calculation is based on an explicit description of a representative from each twist-isoclass. Our method of construction involves studying the eigenspace structure of the elements of the image of the representation and then picking a suitable basis for the underlying vector space.

Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive integer n. Suppose that d, n and D are bounded. Then an isomorphism of A with M_n(K) can be constructed by a polynomial time ff-algorithm. (An ff-algorithm is a deterministic procedure which is allowed to call oracles for factoring integers and factoring univariate polynomials over finite fields.) As a consequence, we obtain a polynomial time ff-algorithm to compute isomorphisms of central simple algebras of bounded degree over K., 15 pages; Theorem 2 and Lemma 8 corrected

We find equations for the higher-dimensional analogue of the modular curve X 0 ( 3 ) using Mumford's algebraic formalism of algebraic theta functions. As a consequence, we derive a method for the construction of genus 2 hyperelliptic curves over small degree number fields whose Jacobian has complex multiplication and good ordinary reduction at the prime 3. We prove the existence of a quasi-quadratic time algorithm for computing a canonical lift in characteristic 3 based on these equations, with a detailed description of our method in genus 1 and 2.

The groups in this paper are abelian. Let G be a reduced torsion-free finite rank group. Then G is cocommutative if End ( G ) is commutative modulo the nil radical. The class number of G, h ( G ) , is the number of isomorphism classes of groups H that are locally isomorphic ( = nearly isomorphic) to G. We say that G satisfies the power cancellation property if G n ≅ H n for some group H and integer n > 0 implies that G ≅ H . We say that G has a Σ-unique decomposition if G n has a unique direct sum decomposition for each integer n > 0 . Let P o ( G ) = { groups H | H ⊕ H ′ = G m for some group H ′ and some integer m > 0 } . We say that G has internal cancellation if given H , K , L ∈ P o ( G ) such that H ⊕ K ≅ H ⊕ L then K ≅ L . We use the class number to study the torsion-free finite rank groups G that have the power cancellation property, or a Σ-unique decompositions, or the internal cancellation property. Furthermore, we show that the power cancellation property for cocommutative strongly indecomposable reduced torsion-free finite rank groups is equivalent to the problem of determining the class number of an algebraic number field. Let G ¯ be the integral closure of G. Using the Mayer–Vietoris sequence we show that there are finite groups associated with G of orders L ( p ) , m ˆ p , and n ˆ p such that h ( G ) = h ( G ¯ ) m ˆ p L ( p ) n ˆ p . Let E ( p ) = Z + p E ¯ where p is a rational prime and E ¯ is the ring of algebraic integers in the algebraic number field k = Q E ¯ . Let G ( p ) be a group such that End ( G ( p ) ) = E ( p ) . We show that the sequence { L ( p ) h ( G ( p ) ) / h ( G ¯ ( p ) ) | primes p } is asymptotically equal to the sequence { p f − 1 | primes p } where f = [ k : Q ] . Furthermore, for quadratic number fields k, h ( k ) = 1 iff { L ( p ) h ( G ( p ) ) | primes p } is asymptotically equal to the sequence of rational primes. This connects unique factorization in number fields with the sequence of rational primes, and with direct sum properties of integrally closed cocommutative strongly indecomposable torsion-free finite rank groups.

Let k be a subfield of a p-adic field of odd residue characteristic, and let L be the function field of a variety of dimension n >= 1 over k. Then Hilbert's Tenth Problem for L is undecidable. In particular, Hilbert's Tenth Problem for function fields of varieties over number fields of dimension >= 1 is undecidable., 19 pages; to appear in Journal of Algebra

In this paper, we always assume that F = Q ( d ) and E = Q ( − d ) , d a squarefree integer, are quadratic number fields with d = u 2 − 2 w 2 , u , w ∈ N . This paper is mainly to give the formula: 8-rank of K 2 O F = 8 -rank of C ( E ) + a ( F ) + σ , where C ( E ) is the narrow class group of E, a ( F ) ∈ { − 1 , 0 , 1 } and σ ∈ { 0 , 1 } ; moreover |8-rank of K 2 O F -8-rank of C ( E ) | ⩽ 1 . This paper is also to show the relations among { − 1 , u + d } ∈ K 2 O F 4 , the dyadic ideal class of C ( E ) and the dyadic ideal class of C ( E ′ ) for E ′ = Q ( − 2 d ) .

Let F be a quadratic number field. We give a criterion, via Hilbert symbols, for an element of order two in the tame kernel of F to be a fourth power in the tame kernel of F. The result can be applied to compute the 8-rank of the tame kernel of F and the Tate kernel of an imaginary quadratic number field. We list the 8-ranks of K 2 O F for all quadratic number fields whose discriminants have exactly two odd prime divisors. In the case when F is an imaginary quadratic number field with the 8-rank of K 2 O F = 0 , the Tate kernel of F is given too. An application of our method to the maximal real subfield of a cyclotomic field is discussed. Numerical examples, in particular the examples of quadratic number fields F with 4-rank of K 2 O F = 8-rank of K 2 O F = 2 illustrate our results.

If G is a simply connected semisimple group defined over a number field k and ∞ is the set of all infinite places of k, then G has strong approximation with respect to ∞ if and only if the archimedean part of any k-simple component of the adele group G A is non-compact. Using the affine Bruhat–Tits building, the authors of [W.K. Chan, J. Hsia, On almost strong approximation of algebraic groups, J. Algebra 254 (2002) 441] formulated an almost strong approximation property (ASAP) for groups of compact type, and they proved that ASAP holds for all classical groups of compact type whose Tits indices over k are not 2 A n (d) with d⩾3. In this paper, we show that ASAP holds for groups of types 3,6 D 4 ,G 2 ,F 4 ,E 7 , or E8.

An algebraic integer is called an epsilon-Pisot number (epsilon > 0) if its Galois conjugates have absolute value less then epsilon. Let K be any real algebraic number field. We prove that the subset of K consisting of epsilon-Pisot numbers which have the same degree as that of the field is relatively dense in the real line R. This has some applications to non-stationary products of random matrices involving Salem numbers. (C) 2004 Elsevier Inc. All rights reserved.

Let F be either an algebraic number field or a p -adic field and A a central simple algebra over F . Suppose A is spanned by a multiplicative semigroup Γ ⊂ A with the property that the minimal polynomial of every g ∈ Γ splits over F . Then A represents the trivial class in the Brauer group of F .

The aim of this paper is to give a characterisation of the determinants and signatures of integral ideal lattices over a given algebraic number field. This is then used to obtain an existence criterion for automorphisms of given characteristic polynomial. In particular, we give a different proof of a result of B. Gross and C. McMullen [in preparation].

We study the level of nonformally real function fields of surfaces over number fields and show that it is at most 4 for a large class of surfaces.

In 1991 Dixon and Kovacs 8 showed that for each field K which has finite degree over its prime subfield there is a number d such that every K finite nilpotent irreducible linear group of degree n 2 over K can be ' generated by d n log n elements. Afterwards Bryant et al. 3 proved K that the same is true for solvable linear groups and this led to asking whether a similar result could hold also removing the solvability hypothesis. In 15 it is proved that the answer is positive in the particular case of finite fields. In the present paper we are able to deal with the case of number fields, thus giving a complete solution to the problem. Namely, Ž . denoting by d G the number of generators of a group G, we prove

The notion of a G-function was first introduced by C. L. Siegel in 1929. Later work of Bombieri, Chudnovsky, Andre, and Dwork clarified the Ž . geometric content of that one variable notion, as a solution of a special Ž . type of linear differential operator of arithmetic type or G-operator . A geometric theory of G-functions was established in full generality by Andre and Baldassarri in AB . Ž . We recall a variation of the classical definition. Let K be a number field and let V be its ring of integers. A G-function at the origin defined K o er K is a formal power series

Let p be a prime number and K a number field containing a primitive p-th root of unity. It is known that an unramified cyclic extension L/K of degree p has a power integral basis if it has a normal integral basis. We show that for all p, the converse is not true in general.

In this paper, we will study a family of quintic hypersurfaces and its Abel–Jacobi images. We will construct codimension 2 algebraic cycles on the blowing-up of certain dimension 3 quintic hypersurfaces defined over number fields, and show that they are homologically trivial but have nontrivial images under the 3-adic Abel–Jacobi map.

Letl > 2 be a prime number. Let O Fdenote a ring of algebraic integers of a number fieldFand letGbe the cyclic group of orderl. Consider the ring of integers O Las a locally free O K[G]-module whereL/Kis a tamely ramified Galois extension of number fields with Galois group isomorphic toG. The classes of rings of integers obtained in such a way for a fixed number fieldKcontaining thelth roots of unity form the subgroup of realisable classesRin the locally free class groupCl( O K[G]). On the other hand, one may also look at the Swan subgroupTofCl( O K[G]), formed by the classes of locally free O K[G]-ideals (s, Σ) wheresin O Kis relatively prime toland Σ denotes the sum of the elements ofG. We show thatT(l − 1)/2 ⊂ R ∩ D, whereDis the kernel group ofCl( O K[G]). We also determine necessary and sufficient conditions for when a realisable class is a Swan class. Last, we show thatR ∩ Dis nontrivial forK = Q(ζl) whenl > 3 by providing a nontrivial lower bound for the size of the Swan subgroupT.

Let us call a profinite group D adelic, if D is a subgroup of the product Ž . Ł SL Z for some n G 2. One of the main cases of interest is when D p n p is the profinite completion of an S-arithmetic subgroup of a semisimple algebraic group. Let k be an algebraic number field, S be a finite set of places of k containing the infinite ones, G be a semisimple, connected algebraic group over k, and G be an S-arithmetic group. We say that G satisfies the congruence subgroup property if the S-congruence kernel SŽ . Ž w x. 4 C G is a finite group see PR2, Chap. 9 . If n G 3, and S s ` , the SŽ . S-congruence kernel C SL is trivial, and so we can identify the profinite n $ Ž . Ž . completion SL Z of SL Z with the product Ł SL Z . Ž . n n p n p The purpose of this note is twofold. Firstly, we prove some results on bounded generation for an arbitrary finitely generated adelic, profinite group. Then, we apply these results to the case of arithmetic groups to prove the following conjecture of A. Lubotzky. ˆ Conjecture. If the profinite completion G of an S-arithmetic group G is adelic, then G satisfies the congruence subgroup property.

Letpbe a prime number and Λ the associated Iwasawa algebra. LetMbe a noetherian Λ-module which defines a representation of a finite Galois groupG. For a given collection of subgroups ofG, we show that relations between idempotents of K[G] yield relations among the degrees of the characteristic power series related to the various submodules of fixed points for each subgroup. This generalizes former results due to Kani (in the case of the genus of algebraic curves), and to Madan and Zimmer (for the classical Iwasawa invariant related to Zp-extensions). The method also gives rise to analogous results for the generalised Selmer group of ap-adic representation associated to the absolute Galois group of a number field. A non-Galois version of these results is also given.Soitpun nombre premier et Λ l'algèbre d'Iwasawa correspondante. On étudie certaines reprèsentations d'un groupe de Galois finiG, associées à des Λ-modules de cotype fini. Une famille de sous-groupes étant donnée, on montre que des relations entre idempotents de l'algèbre Q[G] comduisent à des relations entre les degrés des séries caractéristiques attachées aux sous-modules des points fixes pour chacun de ces sous-groupes. Ceci généralise des résultats antérieurs dus à Kani, pour le genre de courbes algébriques, et à Madan et Zimmer, pour l'invariant lambda classique de Zp-extensions. La méthode décrite fournit aussi l'analogue de ces formules pour le groupe de Selmer généralisé associé à une représentationp-adique du groupe de Galois absolu d'un corps de nombres. On propose aussi une version non-galoisienne de ces résultats.

In this paper, we study Hochschild homology, cyclic homology and K -theory of commutative algebras of finite type over a characteristic zero field. We prove that local complete intersections are characterized by HH i n = 0 for i n / 2 or equivalently by HC i n = 0 for i n / 2. For artinian algebras over a number field, we prove that local complete intersections are characterized by K i n = 0 for i n + 1)/ 2. This last result answers, in the particular case of artinian algebras over a number field, a famous conjecture of Beilinson and Soule about the γ-filtration of the K -theory of a commutative algebra, module torsion.

Let p be an odd prime, and let P denote the class of p-groups which occur as Sylow p-subgroups of finite Galois groups over the p-adic field Qp. We prove that P contains every abelian p-group of rank ≤ (p − 1)2, and that certain nonabelian p-groups do not belong to P . For certain number fields K, we show that P coincides with the class of Sylow p-subgroups of K-admissible groups.

This paper introduces the notions of Diophantine generation and Diophantine equivalence and explores the resulting Diophantine classes of holomorphy rings of number fields and of one variable function fields over finite fields of constants.

Let K be an algebraic number field which contains the d th roots of unity. We will prove that all double covers of the generalized symmetric group Z d ≀ S m are realizable as a Galois group over K and over K ( T ), if d is odd. If d is even, we will determine all double covers of Z d ≀ G m which can be shown to be Galois groups over K and over K ( T ) using Serre′s formula on trace forms. If d ≠ 1 we will use trinomials ƒ( X d ) such that the Galois group of ƒ( X ) = X m + aX l + b ∈ K [ X ] is S m .

Let N/K be a finite normal extension of number fields with G = Gal(N/K). By E. Noether’s theorem the ring of integers ON of N is a projective Z[G]module if and only if N/K is at most tamely ramified. Thus, for such N/K, 0, defines a class Q,(N/K) = (ON) [K : Q](Z[G]) in the class group Cl(Z[G] ), the finite torsion subgroup of the Grothendieck group K,,( Z[G] ) of finitely generated G-modules of finite projective dimension. In [14], M. Taylor proved that in this case, Q,(N/K) is equal to the root number class WNIK, which was defined by Ph. Cassou-Nogds and Friihlich by means of the Artin root numbers of the irreducible symplectic representations of G (cf. [S, Chap. 1 ] or [3, p. 183). In [4], T. Chinburg defined Galois invariants Q(N/K, i), i = 1,2, 3, of N/K in Cl(Z[G]) and proved Q(N/K, 2)=QJN/K) for arbitrary N/K which are at most tamely ramified. Since both classes, Q(N/K, 2) and W,,, are defined for all N/K including those which are wildly ramified, one may be led to the following conjecture.

Let G be a finite group and let p be a fixed prime number. In this paper we shall study the numbers of irreducible characters in a p-block of G with an abelian defect group and the contributions of subsections to the inner product of irreducible characters of G. As is seen in [ 10, 111 Broue and Puig’s generalized characters, which are defined in [3], are relevant for the study of p-blocks with abelian defect groups. They will play an important role in this paper too. Let K be the algebraic number field containing the /Gi th roots of 1, let p be a prime ideal divisor of p in K, and let o be the ring of p-integers in K. Let B be a p-block and let (n, b) be a subsection associated with B. (n, b) is a pair such that ‘II is a p-element of G and b is a p-block of C(n) associated with B. In [3] (n, b) is called a (B, G)-Brauer element. Let x be an ordinary irreducible character in B. The function x”’ ‘) is the central function on G which vanishes outside the p-section of rc and which is such that x(X’ “(np) = x(npE,), where p is a p-regular element of C(n) and E, is the block idempotent of oC(n) corresponding to b. Following [I], for x, XI E B, we denote by m, xs (X3 ‘) the inner product of ~(“9 ‘) and x”“’ ‘) ~ rnpx?) is equal to the inner product of ~(~3 b, and 2’ by Brauer and &ma’s orthogonality relation. rnpxf) is called the contribution of (n, b) to the inner product (x, x’). By [l, (5B)]

A well known theorem of L. Carlitz states that classical algebraic number fields have class number less than or equal to two if and only if any two factorizations of an algebraic integer into irreducible elements contains the same number of factors. We show that the same result holds using a somewhat weaker factorization condition. This leads to a characterization of Dedekind domains with class number less than or equal to three.

We say that a field k possesses local class jield theory when the finite separable algebraic extensions K / k form an Artin-Tate class formation with respect to the multiplicative groups K *. In addition to the p-adic number fields Q, and the field lR of real numbers themselves, their maximal absolutely algebraic subfields Q; and lRa are known to have local class field theory; in fact, being henselian these latter fields represent the equivalents of Q, , lR for almost all number-theoretic considerations, and they have the additional advantage that they may be commonly embedded in the field a of all algebraic numbers. A close examination shows that not only the finite but also certain precisely distinguishable infinite algebraic extensions k of Qz allow local class field theory. In the present note we show that the fields mentioned here are the only subfields of a with local class field theory. If N j k is a finite or infinite Galois field extension we denote the Galois group of N 1 k by G,,, and in particular we write G, = G,;, when N is the separable algebraic closure k of k. The cohomology groups

Let G be a group of finite order g and p be a prime number. Let K be the field which is generated by the gth roots of unity over the p-adic number field, R be the ring of local integers in K and P be the maximal ideal in R. We denote by o the ring R or the residue class field F of R modulo P. We view the group algebra oG as a right o(G x G)-module in the usual way so that y(x, y) = x-‘yy, y E oG and x, y E G. An indecomposable direct summand of the o(G x G)-module oG is the same as a block ideal of oG. Hence, if. 1 =E, +E,+ 0.. +E

In [+6] we proved Conjecture A if K is either an algebraic number field or the completion of an algebraic number field. The proofs given involved extremely technical number theoretic methods and were not applicable to other fields. In this paper we approach this question from a different viewpoint and prove a result valid for arbitrary fields. In the process we prove Conjecture A for a more general class of both K and D, and we obtain greatly simplified proofs of some of the main results of [4-61. By a K-division ring we mean a finite dimensional division algebra over K with center K. In view of Herstein’s result that finite subgroups of division rings of prime characteristic are cyclic [7], we restrict our attention to fields of characteristic zero. We let Q denote the field of rational numbers and, for G a finite multiplicative subgroup of D, we set 9(G) = {C a& ) ai E Q, Ai E G). g’(G) is a finite dimensional division algebra over Q (not necessarily central over Q). Amitsur in [2] determined the structure of g(G); in particular 93(G) depends on G up to isomorphism, and not on D. We denote the center of 93(G) by 8. Since D 3 g(G), D contains the subalgebra generated by K and g(G), which is easily seen to be KC? @ 99(G). We set A(G) = KC? OS S(G). We will maintain this notation throughout this paper.

The simpliest, and most important, class of Krull domains are the factorial domains (UFDs), i.e., those Krull domains with trivial divisor class group. Probably the next simpliest class of Krull domains are the Krull domains with torsion divisor class group, i.e., the almost factorial Krull domains [21]. Almost factorial Krull domains occur very frequently: examples include factorial domains, Krull domains with finite divisor class group, and hence any ring of integers of an algebraic number field. Moreover, Goldman [16, Corollary 21 has shown that any Dedekind domain R has an almost factorial overring A with the same unit group as R. The most important property of almost factorial Krull domains is that they are precisely the Krull domains R such that each subintersection of R is a localization of R. Other characterizations of almost factorial Krull domains may be found in [ 131 or [21]. Almost factorial domains have also been called semifactorial [ 191 and prefactorial. Recently there have been several papers on rings each of whose proper localizations or proper overrings satisfy certain ring-theoretic properties, for example [4,5, 8,9]. Following Fossum [ 13, p. 811, in [4] we defined

Let K be a field. A finite group G is called K-adequate if there exists a K-division ring D (a finite-dimensional division algebra central over K) such that G is contained in D*, the multiplicative group of nonzero elements of D. Fein and Schacher have investigated the problem of determining for which fields K there exists a noncyclic group of odd order which is K-adequate. In [5, 61 they solved this problem when K is an algebraic number field or p-local field. In this paper we do the same for noncyclic groups of even order. Our main result is that if K is an algebraic number field then there exists a noncyclic group of even order which is K-adequate. We show this is false for p-local fields and determine necessary and sufficient conditions on K for there to exist an even-order noncyclic group which is K-adequate. We adopt the notation and terminology of [6]. Recall that for any natural number TZ, cIZ denotes a primitive nth root of unity. If u and ZI are integers j3(u, V) is the highest power of u dividing v and [u, v] is the order of u modulo v. If K is an algebraic number field and y is a prime of K we denote the completion of K at y by K,, . For a finite extension L of K and a prime 7 of L dividing y we denote the relative degree of q over y by f(~/r). If E and L are local fields we denote the ramification degree from L to E by e(E/L). We assume all algebras are finite-dimensional over their centers and all fields are of characteristic zero. We use freely the classification of division algebras over global and local fields by means of Hasse invariants. The reader is referred to [3, Chap. 71 for a discussion of this material.

Let K/k be a Galois extension of number fields with Galois group r, and let D, o be the rings of integers of K, k, respectively. In this paper we investigate the structure of 0 as an or-module by comparing 0 with its dual, Horn@, 0). M. J. Taylor [ 11, Theorem, p. 1731 proved in 1978 that these two modules are stably isomorphic over Zr, assuming that all the primes of o dividing the order of r are unramified in K. This result had been conjectured by Frohlich [5, p. 4231 in the case k = Q. Its generalization to all tame extensions is an easy consequence of Taylor’s subsequent proof of Frohlich’s conjecture describing the class of 0 in the class group of locally free ZTmodules in terms of the Artin root number [ 12, Theorem 1, p. 4 1; 5, Proposition 3, p. 4391. To study the Galois module structure of 0 in the wild case Queyrut [7] introduced the S-Grothendieck groups Gi(oT) and K”(oT), where S is a set of primes of o. Namely, G&(oT) is the Grothendieck group corresponding to the category of finitely generated o-torsion free or-modules, with relations arising from short exact sequences splitting outside S; Ki(oT) is the Grothendieck group of all finitely generated o-torsion free or-modules which are locally projective outside S, with relations arising from short exact sequences. With Cassou-Nogues [3, Corollaire 6.3, p. 231, Queyrut generalized Taylor’s duality result to the wild case by showing that if S is a set of rational primes containing those with a divisor in o wildly ramified in K, then the class of 0 is equal to the class of its dual in K:(U). As in the tame case, this result could be obtained as a consequence of Queyrut’s conjecture concerning the class of 0 in K~(ZI’) [3, p. 81, which was proved in several cases [3]. The proofs of the above results used the description of certain Grothendieck groups as quotients of groups of character functions, as well as results on modules introduced by Swan in [lo].

This paper generalizes a result of Okuyama [4, Theorem 31 on the situation wherein a block of a subgroup of a finite group induces to a block of the group (in the classical sense of Brauer; see [ 1, p. 1361). Our generalization, which comprises Lemma A and Theorem B, is proved mainly by observing that Okuyama’s arguments hold under less stringent hypotheses. Our results are applied in Theorem C to establish a necessary and sufficient condition for when a block of defect zero of a subgroup induces to a block of the group. Another application is Theorem D, which concerns block induction from a normal subgroup. Throughout the paper, H denotes a subgroup of an arbitrary finite group G and p is a fixed prime rational integer. Let R be the ring of integers in a p-adic number field K of characteristic zero, let rt be a generator of J(R), and set R = R/rcR. If M is an RG-module, i@ denotes M/nM. Assume that K and R are splitting fields for all subgroups of G. Let v denote the p-adic valuation on K, scaled so that v(p) = 1. Let v( / G I) = a and r( 1 H 1) = m, so that IGJP=p”, /I1l,,=p”‘. Let B, h denote fixed p-blocks of G (resp. H), where B has defect d Let Irr (B) (resp. Irr(b)), denote the set of ordinary irreducible characters in B (resp. h). If 0 is a rational integral combination of Brauer characters of G, let 8 denote the generalized character of G defined by [ 1, IV.1.21

where p,,, := eZnr “I. Equivalently, NS~( F) = 2 n PC”‘); the product is taken over all prime numbers 1, and n,(F) := max{ n 3 0: E,, G F}, with E,, being the maxima1 totally real subfield of the full cyclotomic field O(n,,,). The work of Mazur and Wiles on the “Main conjecture” confirms the odd part of the BirchhTate conjecture if F is abelian. The 2-part has been established for certain families of abelian number fields, see [3, 4, 5, 6, 73. We give a proof for all totally real-&not necessarily abelian--number fields having the property that 2rlQ1 is the exact 2-power dividing ,v,(F) cp( 1) or the order of Kz(oP), respectively. It is based on results given in [l] and [2].

CONTENTS. I. The ciassical rheoy qf cutx~es orer a finite Jeld. 2. The ‘CFclotomic’ construction for number Jelds and function jields. 2.1. Number fields. 2.2. Function fields. 3. Complex arld p-adic zeta-functions. 3.1. The Complex theory. 3.2. The p-adic theory. 4. Classical Iwasabila rheory. 4.1. Kummer’s criterion. 4.2. The theoq of the tower. 5. Two-variable zera-functions. 5.1. The theory at infinity. 5.2. Special values. 6. The theory for futzction fields. 6.1. The basic set-up. 6.2. The Kummer criterion. 6.3. The theory of the tower. 7. L’nirs. 7. I. Number fields. 7.2. Function fields. References.

It is shown that every central embedding problem E for the absolute Galois group G of a number field has a so-called cyclic reduction E′; this is a central embedding problem for G with a cyclic quotient group J of G such that E is solvable if and only if E′ is solvable. Some information about the minimal order of J is also provided.