Profinite version of the Beckmann-Black problem

Let Q be the rational number field, K/Q be a finite Galois extension with the Galois group G, and let CK be the ideal class group of K in the wider sense. We consider CK as a G-module. Denote by I the augmentation ideal of the group ring of G over the ring of rational integers. Then CK/I(CK) is called the central ideal class group of K, which is the maximal factor group of CK on which G acts trivially. A. Fröhlich [3, 41 rationally determined the central ideal class group of a complete Abelian field over Q whose degree is some power of a prime. The proof is based on Theorems 3 and 4 of Fröhlich [2]. D. Garbanati [6] recently gave an algorithm which will produce the l-invariants of the central ideal class group of an Abelian extension over Q for each prime l dividing its order.

A class of soluble groups

An integral domain $R$ is a \textit{degree-domain} if for given two polynomials $f(x)$ and $g(x)$ in $R[x]$ such that for all $k\in R$ $(g(k)\not=0\Rightarrow g(k)|f(k))$, then $f(x)=0$ or $\deg f\geq \deg g$. We prove that the ring of integers $\mathcal{O}_L$ is a degree-domain, where $\mathbb{Q}\subseteq L$ is a finite Galois extension. Then we study degree-domains that are also unique factorization domains to determine divisibility of polynomials using polynomial evaluations

Let $F$ be a field and let $V$ be a valuation ring in $F$. If $A$ is a central simple $F$-algebra then $V$ can be extended to a Dubrovin valuation ring in $A$. In this paper we consider the structure of Dubrovin valuation rings with center $V$ in crossed product algebras $(K/F,G,f)$ where $K/F$ is a finite Galois extension with Galois group $G$ unramified over $V$ and $f$ is a normalized two-cocycle. In the case where $V$ is indecomposed in $K$ we introduce a family of orders naturally associated to $f$, examine their basic properties, and determine which of these orders is Dubrovin. In the case where $V$ is decomposed we determine the structure in the case of certain special discrete, finite rank valuations.

Let R denote a complete discrete rank one valuation ring of unequal characteristic, and let p denote the characteristic of the residue class field R̅ of R. Consider the integral closure S of R in a finite Galois extension K of the quotient field k of R. Recall (see Prop. 1.1 of [3]) that the inertia group G0 of K over k is a semi-direct product G0 = J × Gp, where J is a cyclic group of order relatively prime to p and Gp is a normal p-subgroup of G.

If K / ℚ {K/\mathbb{Q}} is a finite Galois extension with an almost monomial Galois group and if s 0 ∈ ℂ ∖ { 1 } {s_{0}\in\mathbb{C}\setminus\{1\}} is not a common zero for any two Artin L-functions associated to distinct complex irreducible characters of the Galois group, then all Artin L-functions of K / ℚ {K/\mathbb{Q}} are holomorphic at s 0 {s_{0}} . We present examples and basic properties of almost monomial groups.

Are number fields determined by Artin L-functions?

Let k be an algebraic number field (of finite degree) and K/k be a (finite) Galois extension with Galois group R. In connection with the class field theory, Weil [101 derived a certain canonical factor set a(o, r) of 9 in the idele-class group (K of K, which is uniquely determined by K/k in the sense of equivalence. For instance, a (mod R') -?f t r(o, A) (Ek k, essentially) (mod NK/k( RK)) is the Artin reciprocity law isomoprhism of 9/9' and Sk/NK/k( SK), where 9' is the commutator subgroup of R. A different approach to this (idele-class) factor set was given in [6]. Let, for each pair o, r, a(o, r) be a representative idele of a(o, r). Then (3a)(p, o, r) = a(o, r)a(po, r)-'a(p, o-r)a(p, oY)is, for each triple p, a, r, a principal idele, i.e. a non-zero element of K. We obtain thus a 3-cochain a(p, a, r) = (6a)(p, a, r) of 9 in the principal idele group PK of K, which is in fact a 3-cocycle. We saw, in [6], that this 3-cycocle a, or more precisely its cohomology class, attached in invariant fashion to the Galois extension K/k, has the order m/m', where m = (K: k) and m' is the least common multiple of the p-degrees of K/k, p running over all primes in k. It was shown further, in a joint paper [4] by Hochschild and the writer, that the cyclic group, of order m/m', generated by this 3-cohomology class is exactly the group of all 3-cohomology classes of 9 in PK with ceiling. This last group is in turn, according to EilenbergMacLane [1], nothing but the group of all Teichmtiller 3-cohomology classes for K/k, belonging to central simple algebras over K to which every element of ft can be extended (as automorphism). Now arises the problem to determine the exact algebra-class (though not unique) which possesses our a, attached to K/k, as its Teichmiiller-class. The present work is devoted to this problem and our solution is given in our Theorem in ?3. If a central simple algebra A over K, to which every automorphism of K/k can be extended, is given in a crossed-product form for instance, we can obtain its Teichmiiller-cocycle in a more or less explicit process involving the HilbertSpeiser theorem. On the other hand, in order to obtain our 3-cocycle a, we repeat the construction of the canonical idele-class factor set a given in [6], which is nothing but the global analogy of the local construction in Hochschild [3]1 and makes use of the idele interpretation of the principal genus theorem of Noether, but with more detailed analysis. Namely, we carry out all the processes in terms of ideles, rather than of idele-classes, preserving thus all principal idele factors which were discarded in [6]. This construction, involving the theorem of everywhere splitting algebras, the Hilbert-Speiser theorem and its idble analogy, leads to a. Though fairly explicit, these constructions of a and the Teichmfiller-

Decomposition of integer-valued polynomial algebras

We extend and refine the theory of “organizing modules” of Mazur and Rubin to construct a canonical class of matrices that encodes a range of information about natural families of complexes in arithmetic. We then describe several concrete applications of this theory including the proof of new results on the explicit structures of Galois groups, ideal class groups, and wild kernels in higher algebraic K-theory and the formulation of a range of explicit conjectures concerning both the ranks and Galois structures of Selmer groups of abelian varieties over finite (nonabelian) Galois extensions of number fields.

We now apply the cohomology theory of the previous chapter to the study of the Brauer group. However, we shall have to use a slightly modified construction which takes into account the fact that the absolute Galois group of a field is determined by its finite quotients. This is the cohomology theory of profinite groups, which we develop first. As a fruit of our labours, we identify the Brauer group of a field with a second, this time commutative, cohomology group of the absolute Galois group. As applications, we obtain easy proofs of some basic facts concerning the Brauer group, and give an important characterization of the index of a central simple algebra. Last but not least, one of the main objects of study in this book makes its appearance: the Galois symbol. The cohomology theory of profinite groups was introduced in the late 1950s by John Tate, motivated by sheaf-theoretic considerations of Alexander Grothendieck. His original aim was to find the appropriate formalism for developing class field theory. Tate himself never published his work, which thus became accessible to the larger mathematical community through the famous account of Serre [4], which also contains many original contributions. It was Brauer himself who described the Brauer group as a second cohomology group, using his language of factor systems. We owe to Serre the insight that descent theory can be used to give a more conceptual proof. The Galois symbol was defined by Tate in connection with the algebraic theory of power residue symbols, a topic extensively studied in the 1960s by Bass, Milnor, Moore, Serre and others. Profinite groups and Galois groups It can be no surprise that the main application of the cohomological techniques of the previous chapter will be in the case when G is the Galois group of a finite Galois extension. However, it will be convenient to consider the case of infinite Galois extensions as well, and first and foremost that of the extension k s |k , where k s is a separable closure of k .

In the paper, the structure of the $$ \mathcal{O} $$ K [G]-module F( $$ \mathfrak{m} $$ M ) is described, where M/L, L/K, and K/ℚ p are finite Galois extensions (p is a fixed prime number), G = Gal(M/L), $$ \mathfrak{m} $$ M is a maximal ideal of the ring of integers $$ \mathcal{O} $$ M, and F is a Lubin–Tate formal group law over the ring $$ \mathcal{O} $$ K for a fixed uniformizer π.

In this chapter, we investigate infinite Galois extensions and prove an analog of the fundamental theorem of Galois theory for infinite extensions. The key idea is to put a topology on the Galois group of an infinite dimensional Galois extension and then use this topology to determine which subgroups of the Galois group arise as Galois groups of intermediate extensions. We also give a number of constructions of infinite Galois extensions, constructions that arise in quadratic form theory, number theory, and Galois cohomology, among other places.

Let k be either a local or a global field, and K be a finite Galois extension of k with g = Gal (K/k). Let L be a Galois extension of K which is also Galois over k. Such an extension is called central if Gal(L/iT) lies inside the centre of Gal(L/K). Clearly L is abelian over K. Next set L* = L∩K · kab where kab is the maximal abelian extension of k in its algebraic closure. This is the genus field of L over K/k.

Hume argues for the finite divisibility of space by starting with proving against the infinite divisibility of space. First and foremost, according to Hume, the mind’s capacity is finite so that it is impossible to contain an infinite number of ideas. However, in terms of his concept of general ideas, Hume admits that the mind has the ability to call up an infinite number of ideas, if provoked. Next, through observations of the imagination and the impression, Hume demonstrates that there must be simple and minimal ideas which constitute the general ideas of space and time and are themselves indivisible. The fact that one might be able to perceive a smaller thing with the help of telescopes does not show that our mind is incapable of the smallest ideas, but that the mind could be mistaken to take what is not the smallest as the smallest. One question which is discussed most is whether Hume is legitimate to make the reference from impressions and ideas to things themselves. However, Hume is less concerned with the so-called nature of things and focuses on how things appear to human beings. On the other way, Hume builds his argument upon the existence of minima. For Hume, all ideas are imagistic and extended and if added up ad infinitum, they would result in infinitude. Therefore, no finite extension could be said to be constituted by infinite ideas. Since only what is resolved into units really exists, an infinite extension does not exist at all. This statement maps onto the discussion of the special composition question. Like what general ideas are to Hume, composition is nothing but the arrangement of basic elements for the mereological nihilists.