Decomposition of integer-valued polynomial algebras

In this chapter we shall examine invariants of the Galois module structure on the higher-dimensional algebraic K-groups of rings of algebraic integers in number fields. Special cases of these invariants were given in §§1.3.4-1.3.13. The material of this chapter originated in ([132] Chapter VII) where generalizations of the Chin-burg invariant, Ω0(L/K, 3) of [34] were constructed using algebraic K-groups of integers in dimensions two and three. Independently, Pappas (unpublished) had noticed the same construction. In [132] it was shown, in the totally real case, that the image of this new invariant in the kernel group, D(Z[G(L/K)]) of §2.5.26, is represented in the Horn-description of Theorem 2.5.32 by the value of the Artin L-function at s = -1. The examples described in §§1.3.4-1.3.13 provide further evidence to connect the invariant, Ωr (L/K, 3) constructed from K-groups in dimesions 2r and 2r + 1, with behaviour of the Artin L-series at (or near) s = -r.

A solid mathematical technique for handling uncertainty is offered by the Zadeh-conceived notion of fuzzy sets, provides a convenient point of departure for the construction of a conceptual framework. In this paper, we explore the advantages of applying techniques from fuzzy set theory, paving the way for innovative approaches in the study of fuzzy multiplicative ideal theory. This study investigates that every nonzero fuzzy fractional ideal in a Dedekind domain has a unique factorization into fuzzy prime ideals. We demonstrate a few intermediate findings about the localization of a noetherian domain at its fuzzy fractional ideals. Introducing the new concepts and provide some examples via discrete valuation ring illustrates that a discrete valuation ring is the localization of a noetherian domain at its fuzzy fractional ideals.

The article investigates the classification with precision up to equivalence of involutions in the algebra of upper triangular matrices over the ring of integers of algebraic numbers of quadratic fields.The description of involutions in algebras represents one of the classical problems of ring theory. Standard examples of involutions are transposition in matrix algebra and conjugation in the field of complex numbers and the algebra of quaternions.In the case where the field 𝑃 has a characteristic different from two, a complete description of involutions with precision up to their equivalence in the algebra 𝑇𝑛(𝑃) for any natural number 𝑛 was obtained in [15]. In this work [3] involutions in the algebra of upper triangular matrices over commutative rings are studied. If the ring is a field of characteristic 2 or a Boolean ring, then necessary and sufficient conditions for the finiteness of the number of equivalence classesof involutions were found.This article is a continuation of the work of [3]. In the article [3], in particular, the number of equivalence classes of involutions in the algebras of upper triangular matrices over the ring of integers was found. In this regard, the natural result is the problem of describing involutions with precision up to their equivalence in algebras of upper triangular matrices over the ring of algebraic integers of quadratic fields, to which this work is devoted. In the work, the number of equivalence classes of involutions in such algebras is found and the method of finding representatives in each equivalence class is illustrated with examples. Upon receipt the main results in this work, the apparatus of the theory of Pell’s equations is significantly used.

Let R be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate R . We restrict to the class of prime divisors that dominate R and show that if a collection of such prime divisors is taken below a certain “level,” then the intersection is an almost Dedekind domain having the property that every nonzero ideal can be represented uniquely as an irredundant intersection of powers of maximal ideals.

A logarithmic approach to classgroups of integral group rings

Algorithme explicite pour la recherche du P.G.C.D. dans certains anneaux principaux d'entiers de corps de nombres

Number fields and their rings of integers, which generalize the rational numbers and the integers, are foundational objects in number theory. There are several computer algebra systems and databases concerned with the computational aspects of these. In particular, computing the ring of integers of a given number field is one of the main tasks of computational algebraic number theory. In this paper, we describe a formalization in Lean 4 for certifying such computations. In order to accomplish this, we developed several data types amenable to computation. Moreover, many other underlying mathematical concepts and results had to be formalized, most of which are also of independent interest. These include resultants and discriminants, as well as methods for proving irreducibility of univariate polynomials over finite fields and over the rational numbers. To illustrate the feasibility of our strategy, we formally verified entries from the Number fields section of the L-functions and modular forms database (LMFDB). These concern, for several number fields, the explicitly given integral basis of the ring of integers and the discriminant. To accomplish this, we wrote SageMath code that computes the corresponding certificates and outputs a Lean proof of the statement to be verified.

The Beckmann-Black problem asks whether any given finite Galois extension E/K of group G is the specialization at some point t 0 ∈ P 1 ( K ) of some finite regular Galois extension F / K ( T ) with the same group. In this paper, we study a generalization of this problem for infinite extensions, via the profinite twisting lemma, and apply the latter in many situations, like abelian infinite extensions and the l -universal Frattini cover of an arbitrary finite group.

A generalization of the Conner-Floyd map from complex cobordism to complex /C-theory is constructed for formal A-modules when A is the ring of algebraic integers in a number field or its p-adic completion. This map is employed to study the Adams-Novikov spectral sequence for formal A-modules and to confirm a conjecture of D. Ravenel. 0. Introduction. Let BP be the spectrum representing Brown-Peterson cohomology with respect to a prime p and let E be the Adams summand of complex if-theory with respect to this prime. The BP version of the Conner-Floyd map is a map of spectra BP —+ E which induces a natural equivalence BP, X ®Bp. E» ~ E.X. In particular this induces an isomorphism £* ®Bp. BP»BP®bp.£. ~ E.E and so provides a way of computing the Hopf algebra of stable co-operations for E from those for BP. Using this one can obtain a description of E*E similar to that for K*K contained in [AHS]. The study of BP and the computation of BP* BP are based on a study of formal group law, in particular the p-typical formal group law. In [Rl] Ravenel studied a generalization of this situation where the formal group law is replaced by a formal yl-module where A is the ring of integers in an algebraic number field K or its p-adic completion. The purpose of the present paper is to describe the corresponding generalization of the map (BP», BP* BP) —► (E*,E*E) induced by the Conner-Floyd map, and to compute the generalization of E*E. This is of interest because it provides some information about a conjecture (3.10) made in [Rl]. This conjecture concerned the value of a certain Ext group Exty,t(Va, Va) when K is an extension of the field Qp of p-adic numbers. Here (Va, VaT) is the Hopf algebroid corresponding to the A-typical formal A module. This group was conjectured to be, up to small factor, A/J^,^. Here J^iq_1 is the ideal of A generated by the elements of the form aTM — 1 for units a of A congruent to 1 mod(7r) and (it) is the unique prime ideal in A. We will show, using the generalization of the Conner-Floyd map, that A/J*,^ occurs as E\' in the chromatic spectra sequence for formal ^-modules [Rl, Lemma 2.10] and that the small factor in the conjecture is contributed by the nontriviality of the differential d originating from this group. We will analyze this differential and show that it is nonzero for A Received by the editors June 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55T25; Secondary 55N22, 14L05. ©1987 American Mathematical Society 0002-9947/87 SI.00 + $.25 per page 319 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Let H_{D}(T) denote the Hilbert class polynomial of the imaginary quadratic order of discriminant D . We study the rate of growth of the greatest common divisor of H_{D}(a) and H_{D}(b) as |D| \to \infty for a and b belonging to various Dedekind domains. We also study the modular support problem: if for all but finitely many D every prime ideal dividing H_{D}(a) also divides H_{D}(b) , what can we say about a and b ? If we replace H_{D}(T) by T^{n}-1 and the Dedekind domain is a ring of S -integers in some number field, then these are classical questions that have been investigated by Bugeaud–Corvaja–Zannier, Corvaja–Zannier, and Corrales-Rodrigáñez–Schoof.

Let $H$ be a multiplicatively written monoid. Given $k\in{\bf N}^+$, we denote by $\mathscr U_k$ the set of all $\ell\in{\bf N}^+$ such that $a_1\cdots a_k=b_1\cdots b_\ell$ for some atoms $a_1,\ldots,a_k,b_1,\ldots,b_\ell\in H$. The sets $\mathscr U_k$ are one of the most fundamental invariants studied in the theory of non-unique factorization, and understanding their structure is a basic problem in the field: In particular, it is known that, in many cases of interest, these sets are almost arithmetic progressions with the same difference and bound for all large $k$, namely, $H$ satisfies the Structure Theorem for Unions. The present paper improves the current state of the art on this problem. More precisely, we show that, under mild assumptions on $H$, not only does the Structure Theorem for Unions hold, but there also exists $\mu\in{\bf N}^+$ such that, for every $M\in{\bf N}$, the sequences $$ \bigl((\mathscr U_k-\inf\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} \quad\text{and}\quad \bigl((\sup\mathscr U_k-\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} $$ are $\mu$-periodic from some point on. The result applies, e.g., to (the multiplicative monoid of) all commutative Krull domains (e.g., Dedekind domains) with finite class group; a variety of weakly Krull commutative domains (including all orders in number fields with finite elasticity); some maximal orders in central simple algebras over global fields; and all numerical monoids. Large parts of the proofs are worked out in a "purely additive model", by inquiring into the properties of what we call a subadditive family, i.e., a collection $\mathscr L$ of subsets of $\bf N$ such that, for all $L_1,L_2\in\mathscr L$, there is $L\in\mathscr L$ with $L_1+L_2\subseteq L$.

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

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject.Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the fieldQp of p-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel’s Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields Fp to roots of corresponding polynomials over p-adic fields Qp.Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let K be a finite separable extension of a field F, let R be a Dedekind domain with field of fractions F, and let T be the integral closure of R in K. The question concerns the factorization of an ideal pT in T when p is a nonzero prime ideal in R. If Fp denotes the completion of F with respect to p, the theorem explains how the tensor product K ⊗F Fp splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of pT in T to the splitting of Fp in a complete field in which only one of the prime factors of pT plays a role.Section 7 is a brief aside mentioning additional conclusions one can draw when the extension K/F is a Galois extension.Section 8 applies the main theorem of Section 6 to an analysis of the different of K/F and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V.Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of Fq (X), where Fq is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.KeywordsPrime IdealValuation RingBasic AlgebraDedekind DomainFractional IdealThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A modular architecture for very fast digital signal processing (DSP) elements are presented. The computation is performed over finite rings (or fields) and is able to emulate processing over the integer ring using residue number systems. The computations are restricted to closed operations (ring or field binary operators) with the ability to perform limited scaling operations. Computations naturally defined over finite mathematical systems are also easily implemented using this approach. The technique evolves from the decomposition of each closed calculation using the ring/field associativity property. Linear systolic arrays, formed with multiple elements, each of a single generic form, are used for all calculations. The pipeline cycle is determined from the generic cell and is predicted to be very fast by a critical path analysis. The cells are matched to the VLSI medium, and the resulting array structures are very dense. Examples of DSP applications are given to illustrate the technique, and example cell and array VLSI layouts are presented for a 3- mu m CMOS process. >

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.