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Class Field Theory (CFT) is the main achievement of algebraic number theory of the 20th century. Its reach, beauty and power, stemming from the first steps in algebraic number theory by Gauß, have substantially influenced number theory. Shafarevich wrote: 'Weil was undoubtedly right when he asserted, in the preface to the Russian edition of his book on number theory 1 , that since class field theory pertains to the foundation of mathematics, every mathematician should be as familiar with it as with Galois theory. Moreover, just like Galois theory before it, class field theory was reputed to be very complicated and accessible only to experts ... For class field theory, on the other hand, there is a wide range of essentially different expositions, so that it is not immediately obvious even whether the subject is the same'. 2 Weil's opinion has proved to be quixotic: these days even some number theorists are not familiar with the substance of CFT. This text reviews the enduring process of discovering new branches of CFT and its generalisations. Many of such developments were complicated at their early stages and some were difficult or impossible to understand for their contemporaries. Three main generalisations of CFT and their further extensions will be presented and some of their key fundamental features will be discussed. This text proposes eight fundamental problems. We start with Kummer theory, a purely algebraic exercise, whose highly non-trivial arithmetic analogues over arithmetic fields are supplied by CFT. Kummer theory is an algebraic predecessor of CFT including its existence theorem. Then we discuss the fundamental split of (one-dimensional) CFT into special CFT (SCFT) and general CFT (GCFT). This split has enormously affected many developments in number theory. Section 3 delves into four fundamental parts of CFT including the reciprocity map, existence theorem, explicit formulas for the Hilbert symbol and its generalisations, and interaction with ramification theory. Section 4 briefly touches on higher Kummer theory using Milnor K-groups, i.e. the norm residue isomorphism property. Three generalisations of CFT: Langlands correspondences (LC), higher CFT, and anabelian geometry are discussed in section 5. We note that the split of CFT into SCFT and GCFT is currently somehow reproduced at the level of generalisations of CFT: LC over number fields does not yet have any development parallel to GCFT, while higher CFT is parallel to GCFT and it does not have substantial developments similar to SCFT. In the last section we specialise to elliptic curves over global fields, as an illustration. There we consider two further developments: Mochizuki's inter-universal Teichmüller theory (IUT) which is pivoted on anabelian geometry and two-dimensional adelic analysis and geometry which uses structures of two-dimensional CFT. We also consider the fundamental role of zeta integrals which may unite different generalisations of CFT. Similarly to the situation with LC, the current studies of special values of zeta-and L-functions of elliptic curves over number fields, except two-dimensional adelic analysis and geometry, use special structures and are not of general type. There is no attempt to mention all the main results in CFT and all of its generalisations or all of their parts, and the text does not include all of bibliographical references. 1 [64] 2 in Foreword to [13].

Let K = ℚ(θ) be an extension of the field ℚ of rational numbers where θ satisfies an irreducible polynomial xp − a of prime degree belonging to ℤ[x]. In this paper, we give explicilty an integral basis for K using only elementary algebraic number theory. Though an integral basis for such fields is already known (see [Trans. Amer. Math. Soc., 11 (1910), 388–392)], our description of integral basis is different and slightly simpler. We also give a short proof of the formula for discriminant of such fields.

We show that for any integer $N$, there are only finitely many cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,1,...,23\}$. More generally, we define a simple sequence $(r(w))_{w \geq 0}$ such that for any integer $w$, any number field $E$ whose root-discriminant is less than $r(w)$, and any ideal $N$ in the ring of integers of $E$, there are only finitely many cuspidal algebraic automorphic representations of general linear groups over $E$ whose conductor is $N$ and whose weights are in the interval $\{0,1,...,w\}$. Assuming a version of GRH, we also show that we may replace $r(w)$ with $8 \pi e^{\gamma-H_w}$ in this statement, where $\gamma$ is Euler's constant and $H_w$ the $w$-th harmonic number. The proofs are based on some new positivity properties of certain real quadratic forms which occur in the study of the Weil explicit formula for Rankin-Selberg $L$-functions. Both the effectiveness and the optimality of the methods are discussed., Comment: 30 pages, 1 table

Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra-Lenstra-Lov\'asz reduction algorithm (so-called LLL) has been improved in many ways through the past decades and remains one of the central methods used for reducing integral lattice basis. In particular, its floating-point variants-where the rational arithmetic required by Gram-Schmidt orthogonalization is replaced by floating-point arithmetic-are now the fastest known. However, the systematic study of the reduction theory of real quadratic forms or, more generally, of real lattices is not widely represented in the literature. When the problem arises, the lattice is usually replaced by an integral approximation of (a multiple of) the original lattice, which is then reduced. While practically useful and proven in some special cases, this method doesn't offer any guarantee of success in general. In this work, we present an adaptive-precision version of a generalized LLL algorithm that covers this case in all generality. In particular, we replace floating-point arithmetic by Interval Arithmetic to certify the behavior of the algorithm. We conclude by giving a typical application of the result in algebraic number theory for the reduction of ideal lattices in number fields., Comment: 23 pages

Determination of class number of a quadratic number field is one of the fundamental problems in algebraic number theory. This topic have been the object of attention for many years and there exist a large number of interesting results. This is a survey aimed at reviewing some results concerning the criteria for the class number of certain Richaud–Degert type real quadratic number fields to be at most 3.

Finite fields form an important chapter in abstract algebra, and mathematics in general. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a much larger audience. Such lattice models of finite fields provide a good basis for later developing the theory in a more concrete way, including Frobenius elements, all the way to Artin reciprocity law. Examples are provided, intended for an undergraduate audience in the first place., 16 pages, research project

In this work the unsolvability of certain equations is studied in the case of cyclotomic number fields whose ring of integers is not a principal ideal domain. Winterhof et al. considered the equations for \(\gamma \in {\mathbb Z}\). We first extend this result to \(\gamma \in {\mathbb R}\cap {\mathbb Z}[\zeta _m]\) by using a new method from algebraic number theory. Then we present its applications to \(\gamma \)-Butson-Hadamard matrices, \(\gamma \)-Conference matrices and type \(\gamma \) nearly perfect sequences for \(\gamma \in {\mathbb R}\cap {\mathbb Z}[\zeta _m] \).

Let p be an odd prime and consider the finite field F p 2 . Given a linear code C ⊂ F p 2 n , we use algebraic number theory to construct an associated lattice Λ C ⊂ O L n for L an algebraic number field and O L the ring of integers of L. We attach a theta series θ Λ C to the lattice Λ C and prove a relation between θ Λ C and the complete weight enumerator evaluated on weight one theta series.

This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic number theory, (3) modularity. As to (1), we prove that the universal deformation is dihedral if all infinitesimal deformations are dihedral. Concerning (2) in the setting of Galois representations of number fields, we give sufficient conditions to ensure that the universal deformation relatively unramified outside a finite set of primes is dihedral, and discuss in how far these conditions are necessary. As side-results, we obtain cases of the unramified Fontaine-Mazur conjecture, and in many cases positively answer a question of Greenberg and Coleman on the splitting behaviour at p of p-adic Galois representations attached to newforms. As to (3), we prove a modularity theorem of the form `R=T' for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral., 43 pages; minor corrections and improvements following referee's comments

Part 1 Fields with valuations: absolute values the topology defined by an absolute value complete fields valuations, valuation rings and places the representation by power series ordered groups general valuations. Part 2 Extensions: generalities on extensions extensions of complete fields extensions of incomplete fields Dedekind domains and the string approximation theorem extensions of Dedekind domains different and discriminant. Part 3 Global fields: algebraic number fields the product formula the unit theorem the class number. Part 4 Function fields: divisors on a function field principal divisors and the divisor class group Riemann's theorem and the speciality index the genus derivations and differentials the Riemann-Roch theorem and its consequences elliptic function fields Abelian integrals and the Abel-Jacobi theorem. Part 5 Algebraic function fields in two variables: valuations on function fields of two variables.

A Hadamard matrix is a square matrix with entries \(\pm 1\) whose rows are orthogonal to each other. Hadamard matrices appear in various fields including cryptography, coding theory, combinatorics etc. This study takes an interest in \(\gamma \) near Butson-Hadamard matrix that is a generalization of Hadamard matrices for \(\gamma \in {\mathbb R}\cap {\mathbb Z}[\zeta _m] \). These matrices are examined in this study. In particular, the unsolvability of certain equations is studied in the case of cyclotomic number fields. Winterhof et al. considered the equations for \(\gamma \in {\mathbb Z}\), and by the authors for \(\gamma \in {\mathbb R}\cap {\mathbb Z}[\zeta _m]\). In this study, we obtain another method for checking the nonexistence cases of these equations, which uses the tool of norm from algebraic number theory. Then, the direct applications of these results to \(\gamma \) near Butson-Hadamard matrices are obtained. In the second part of this study, the connection between nonlinear Boolean cryptographic functions and \(\gamma \) near Butson-Hadamard matrices having small \(|\gamma |\) is established. In addition, a computer search is done for checking the cases which are excluded by our results and for obtaining new examples of existence parameters.

Let O be the ring of S-integers in a number field k. We prove that if the group of units O^* is infinite then every matrix in $\Gamma$ = SL_2(O) is a product of at most 9 elementary matrices. This completes a long line of research in this direction. As a consequence, we obtain that $\Gamma$ is boundedly generated as an abstract group., Comment: Final version - to appear in `Algebra and Number Theory'

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 .

An integral self-affine tile is the solution of a set equation A T = ⋃ d ∈ D ( T + d ) \mathbf {A} \mathcal {T} = \bigcup _{d \in \mathcal {D}} (\mathcal {T} + d) , where A \mathbf {A} is an n × n n \times n integer matrix and D \mathcal {D} is a finite subset of Z n \mathbb {Z}^n . In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices A ∈ Q n × n \mathbf {A} \in \mathbb {Q}^{n \times n} . We define rational self-affine tiles as compact subsets of the open subring R n × ∏ p K p \mathbb {R}^n\times \prod _\mathfrak {p} K_\mathfrak {p} of the adèle ring A K \mathbb {A}_K , where the factors of the (finite) product are certain p \mathfrak {p} -adic completions of a number field K K that is defined in terms of the characteristic polynomial of A \mathbf {A} . Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tile with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with R n × ∏ p { 0 } ≃ R n \mathbb {R}^n \times \prod _\mathfrak {p} \{0\} \simeq \mathbb {R}^n . Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of the digit set D \mathcal {D} , intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. This enables us to gain new results for tilings associated with numeration systems.

In the recent PhD thesis of Bouw, an algorithm is examined that computes the group structure of the principal units of a p-adic number field completion. In the same thesis, this algorithm is used to compute Hilbert norm residue symbols. In the present paper, we will demonstrate two other applications. The first application is the computation of an important invariant of number field completions, called ibeta. The algorithm that computes ibeta is deterministic and runs in polynomial time. The second application uses Hilbert norm residue symbols and yields a probabilistic algorithm that computes the m-th power residue symbol (a/b)m in arbitrary number fields K. This probabilistic algorithm relies on LLL-reduction and sampling of nearprimes. Using heuristics, we analyse its complexity to be polynomial expected time in n = [K: ℚ], log |ΔK| and the input bit size - a tentative conclusion corroborated by timing experiments. An implementation of the algorithm in Magma will be available at https://github.com/kodebro/powerresiduesymbol.

Finding a short elementggof a number field, given the ideal generated bygg, is a classic problem in computational algebraic number theory. Solving this problem recovers the private key in cryptosystems introduced by Gentry, Smart-Vercauteren, Gentry-Halevi, Garg-Gentry-Halevi, et al. Work over the last few years has shown that for some number fields this problem has a surprisingly low post-quantum security level. This paper shows, and experimentally verifies, that for some number fields this problem has a surprisingly low pre-quantum security level

Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs and their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gr\"obner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases where there are additional symmetries). Here we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of a SIC. Using this method we have calculated 69 new exact solutions, including 9 new dimensions where previously only numerical solutions were known, which more than triples the number of known exact solutions. In some cases the solutions require number fields with degrees as high as 12,288. We use these solutions to confirm that they obey the number-theoretic conjectures and we address two questions suggested by the previous work., Comment: 22 pages + 19 page appendix with many data tables. v2: published version

We develop an efficient algorithm to solve Mordellʼs equation over global function fields. Our method involves ideas from algebraic number theory, especially class field theory. For explicit calculations we used Magma and KASH. Contrary to the number field case the number of solutions can be infinite.

This chapter establishes some essential foundational results in the subject of algebraic number theory beyond what was already in Basic Algebra. Section 1 puts matters in perspective by examining what was proved in Chapter I for quadratic number fields and picking out questions that need to be addressed before one can hope to develop a comparable theory for number fields of degree greater than 2. Sections 2–4 concern the field discriminant of a number field. Section 2 contains the definition of discriminant, as well as some formulas and examples. The main result of Section 3 is the Dedekind Discriminant Theorem. This concerns how prime ideals $(p)$ in $\mathbb{Z}$ split when extended to the ideal $(p)R$ in the ring of integers $R$ of a number field. The theorem says that ramification, i.e, the occurrence of some prime ideal factor in $R$ to a power greater than 1, occurs if and only if $p$ divides the field discriminant. The theorem is proved only in a very useful special case, the general case being deferred to Chapter VI. The useful special case is obtained as a consequence of Kummer's criterion, which relates the factorization modulo $p$ of irreducible monic polynomials in $\mathbb{Z}[X]$ to the question of the splitting of the ideal $(p)R$. Section 4 gives a number of examples of the theory for number fields of degree 3. Section 5 establishes the Dirichlet Unit Theorem, which describes the group of units in the ring of algebraic integers in a number field. The torsion subgroup is the subgroup of roots of unity, and it is finite. The quotient of the group of units by the torsion subgroup is a free abelian group of a certain finite rank. The proof is an application of the Minkowski Lattice-Point Theorem. Section 6 concerns class numbers of algebraic number fields. Two nonzero ideals $I$ and $J$ in the ring of algebraic integers of a number field are equivalent if there are nonzero principal ideals $(a)$ and $(b)$ with $(a)I=(b)J$. It is relatively easy to prove that the set of equivalence classes has a group structure and that the order of this group, which is called the class number, is finite. The class number is 1 if and only if the ring is a principal ideal domain. One wants to be able to compute class numbers, and this easy proof of finiteness of class numbers is not helpful toward this end. Instead, one applies the Minkowski Lattice-Point Theorem a second time, obtaining a second proof of finiteness, one that has a sharp estimate for a finite set of ideals that need to be tested for equivalence. Some examples are provided. A by-product of the sharp estimate is Minkowski's theorem that the field discriminant of any number field other than $\mathbb{Q}$ is greater than 1. In combination with the Dedekind Discriminant Theorem, this result shows that there always exist ramified primes over $\mathbb{Q}$.

At the end of Sect. 4.6 of Chap. 4, we left ourselves with the problem of determining the finite nonempty subsets S of the positive integers such that for infinitely many primes p, S is a set of non-residues of p. We observed there that if S has this property then the product of all the elements in every subset of S of odd cardinality is never a square. The object of this chapter is to prove the converse of this statement, i.e., we wish to prove Theorem 4.12. The proof of Theorem 4.12 that we present uses ideas that are closely related to the ones that Dirichlet used in his proof of Theorem 4.5, together with some technical improvements due to Hilbert. The key tool that we need is an analytic function attached to algebraic number fields, called the zeta function of the field. The definition of this function requires a significant amount of mathematical technology from the theory of algebraic numbers, and so in Sect. 5.1 we begin with a discussion of the results from algebraic number theory that will be required, with Dedekind’s Ideal Distribution Theorem as the final goal of this section. The zeta function of an algebraic number field is defined and studied in Sect. 5.2; in particular, the Euler-Dedekind product formula for the zeta function is derived here. In Sect. 5.3 a product formula for the zeta function of a quadratic number field that will be required in the proof of Theorem 4.12 is derived from the Euler-Dedekind product formula. The proof of Theorem 4.12, the principal object of this chapter, is carried out in Sect. 5.4 and some results which are closely related to that theorem are also established there. In the interest of completeness, we prove in Sect. 5.5 the Fundamental Theorem of Ideal Theory, Theorem 3.16 of Chap. 3, since it is used in an essential way in the derivation of the Euler-Dedekind product formula.

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 field $\mathbb{Q}_p$ 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 $\mathbb{F}_p$ to roots of corresponding polynomials over $p$-adic fields $\mathbb{Q}_p$. 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 $\mathfrak{p} T$ in $T$ when $\mathfrak{p}$ is a nonzero prime ideal in $R$. If $F_{\mathfrak{p}}$ denotes the completion of $F$ with respect to $\mathfrak{p}$, the theorem explains how the tensor product $K\otimes_FF_{\mathfrak{p}}$ splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of $\mathfrak{p} T$ in $T$ to the splitting of $F_{\mathfrak{p}}$ in a complete field in which only one of the prime factors of $\mathfrak{p} T$ 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 $\mathbb{F}_q(X)$, where $\mathbb{F}_q$ 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.

For a finite abelian p-group A of rank d = dim A/pA, let A := be its (logarithmic) mean exponent. We study the behavior of themean exponent of p-class groups in pro-p towers L/K of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-p towers in which the mean exponent of p-class groups remains bounded. Several explicit examples are given with p = 2. Turning to group theory, we introduce an invariant attached to a finitely generated pro-p group G; when G = Gal(L/K), where L is the Hilbert p-class field tower of a number field K, measures the asymptotic behavior of the mean exponent of p-class groups inside L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

Let $${f(x)=(x-a_1)\cdots (x-a_m)}$$ , where a 1, . . . , a m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether $${(f(x))^{2^k}+1}$$ is irreducible for every k ≥ 1. In 1919 Polya proved that if $${P(x)\in\mathbb{Z}[x]}$$ is of degree m and there are m rational integer values a for which 0

For any odd prime p , we present a reflection theorem which for p = 3 is the Scholz Reflection Theorem. We obtain some formulas for the p -rank and the p n -rank( K 2 O F ), where F is a number field. With the help of our reflection theorem, we establish relations between the p n -divisibility of the order of K 2 O F and the p n -divisibility of the class number of some algebraic number fields F .

Text By a result of G. Freiman and A. Geroldinger [G. Freiman, A. Geroldinger, An addition theorem and its arithmetical application, J. Number Theory 85 (1) (2000) 59–73] it is known that the set of lengths of factorizations of an algebraic integer (in the ring of integers of an algebraic number field), or more generally of an element of a Krull monoid with finite class group, has a certain structure: it is an almost arithmetical multiprogression for whose difference and bound only finitely many values are possible, and these depend just on the class group. We establish a sort of converse to this result, showing that for each choice of finitely many differences and of a bound there exists some number field such that each almost arithmetical multiprogression with one of these difference and that bound is up to shift the set of lengths of an algebraic integer of that number field. Moreover, we give an explicit sufficient condition on the class group of the number field for this to happen. Video For a video summary of this paper, please visit http://www.youtube.com/watch?v=c61xM-5D6Do .

We develop a formalism for studying the discrete logarithm problem for the multiplicative group and for elliptic curves over finite fields by lifting the respective group to an algebraic number field and using global duality. One of our main tools is the signature of a Dirichlet character (in the multiplicative group case) or principal homogeneous space (in the elliptic curve case), which is a measure of its ramification at certain places. We then develop signature calculus, which generalizes and refines the index calculus method. Finally, using some heuristics, we show the random polynomial time equivalence for these two cases between the problem of computing signatures and the discrete logarithm problem. This relates the discrete logarithm problem to some very well-known problems in algebraic number theory and arithmetic geometry.

We study large values of the remainder term in the asymptotic formula for the number of irreducible integers in an algebraic number field K. In the case when the class number h of K is larger than 1, under certain technical condition on multiplicities of non-trivial zeros of Hecke L-functions, we detect oscillations larger than what one could expect on the basis of the classical Littlewood’s omega estimate for the remainder term in the prime number formula. In some cases the main result is unconditional. It is proved that this is always the case when h = 2.

We prove that the the kernel of the reciprocity map for a product of curves over a $p$-adic field with split semi-stable reduction is divisible. We also consider the $K_1$ of a product of curves over a number field., Comment: 14 pages

Let k be an algebraic number field and let N ( k , C l ; m ) denote the number of abelian extensions K of k with G ( K / k ) ≅ C l , the cyclic group of prime order l, and the relative discriminant D ( K / k ) of norm equal to m. In this paper, we derive an asymptotic formula for ∑ m ⩽ X N ( k , C l ; m ) using the class field theory and a method, developed by Wright. We show that our result is identical to a result of Cohen, Diaz y Diaz and Olivier, obtained by methods of classical algebraic number theory, although our methods allow for a more elegant treatment and reduce a global calculation to a series of local calculations.

We give a simple proof of the Siegel-Tatuzawa theorem according to which the residues at s = 1 of the Dedekind zeta functions of quadratic number fields are effectively not too small, with at most one exceptional quadratic field. We then give a simple proof of the Brauer-Siegel theorem for normal number fields which gives the asymptotics for the logarithm of the product of the class number and the regulator of number fields.

Recently the authors showed that the algebraic integers of the form $-m+\zeta_k$ are bases of a canonical number system of $\mathbb{Z}[\zeta_k]$ provided $m\geq \phi(k)+1$, where $\zeta_k$ denotes a $k$-th primitive root of unity and $\phi$ is Euler's totient function. In this paper we are interested in the questions whether two bases $-m+\zeta_k$ and $-n+\zeta_k$ are multiplicatively independent. We show the multiplicative independence in case that $0 1$., Comment: 9 pages

While most algebra is done by writing text and for- mulas, diagrams have always been used to present structural infor- mation clearly and concisely. Text shells are the de facto interface for computational algebraic number theory, but they are inca- pable of presenting structural information graphically. We present GiANT, a newly developed graphical interface for working with number fields. GiANT oers interactive diagrams, drag-and-drop functionality, and typeset formulas.

A new approach is presented for the design of full modulation diversity (FMD) complex lattices for the Rayleigh-fading channel. The FMD lattice design problem essentially consists of maximizing a parameter called the normalized minimum product distance d/sub p//sup 2/ of the finite signal set carved out of the lattice. We approach the problem of maximizing d/sub p//sup 2/ by minimizing the average energy of the signal constellation obtained from a new family of FMD lattices. The unnormalized minimum product distance for every lattice in the proposed family is lower-bounded by a nonzero constant. Minimizing the average energy of the signal set translates to minimizing the Frobenius norm of the generator matrices within the proposed family. The two strategies proposed for the Frobenius norm reduction are based on the concepts of successive minima (SM) and basis reduction of an equivalent real lattice. The lattice constructions in this paper provide significantly larger normalized minimum product distances compared to the existing lattices in certain dimensions. The proposed construction is general and works for any dimension as long as a list of number fields of the same degree is available.

In this paper, we describe an algorithm that reduces the computation of the (full) p p -Selmer group of an elliptic curve E E over a number field to standard number field computations such as determining the ( p p -torsion of) the S S -class group and a basis of the S S -units modulo p p th powers for a suitable set S S of primes. In particular, we give a result reducing this set S S of ‘bad primes’ to a very small set, which in many cases only contains the primes above p p . As of today, this provides a feasible algorithm for performing a full 3 3 -descent on an elliptic curve over Q \mathbb Q , but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of E [ p ] E[p] is favorable, simplifications are possible and p p -descents for larger p p are accessible even today. To demonstrate how the method works, several worked examples are included.

The concept of number systems in higher-dimensional Euclidean spaces as well as in number fields are introduced and treated. The paper is mainly a survey of known results. If ofZ"k denotes the ring of integer vectorials in Euclidean space R"k then we study the subgroup L=MZ"k where M is a k x k matrix with integer entries. A digit set is defined with the help of residue classes (modL)A"j (j = 0,..., t - 1), where t = |det M|. The analogous of q-additive and q-multiplicative functions over the ring of Gaussian integers are also given.

Using degree 2 polynomial formal groups, we construct Hopf algebras over valuation rings of local number fields K that are orders in KG, G cyclic of order p n . The construction does not yield all Greither orders when n = 2, but does yield new classes of Hopf orders for all n ≥= 3, and shows that a general Hopf order in KG, G cyclic of order p n , involves at least n ( n +1)/2 parameters.

Let L be a Galois extension of the number field K. Set n = nK = deg K/ℚ, nL = deg L/ℚ and nL/K = deg L/K. Let I = IL/K denote the group of fractional ideals of K whose prime decomposition contains no prime ideals that ramify in L, and let P = {(α)ΣI: αΣK*, α>0}. Following Hecke [9}, let (λ1, λ2, …, λn − 1) be a basis for the torsion-free characters on P that satisfy λi(α) = 1 (1≤i≤n − 1) for all units α>0 in , the ring of integers of K. Fixing an extension of each λi to a character on I, then λi,(α) (1 ≤i≤n − 1) are defined for all ideals α of K that do not ramify in L. So, for such ideals, we can define . Then the small region of K referred to above isfor 0

In the previous chapters we have worked essentially with ‘classically’ integral solutions, that is over ℤ. However, since the times of Kummer (and even of Gauss) it has been recognized that diophantine equations are most advantageously dealt with by going out of ℚand using tools from Algebraic Number Theory; this also led to consider solutions in integers of number fields, and even in S-integers therein, i.e. those which have a denominator composed only of primes in the finite set S. In turn, new concepts have been created for studying these more general solutions.

In this chapter, we will prove two fundamental results in algebraic number theory: the finiteness of the class group, and Dirichlet’s Unit TheoremDirichlet, Peter Gustav Lejeune!Dirichlet’s unit theorem, which gives the structure of the group of units in the rings of integers of number fields.

It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric familiy of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $\xi$ of the polynomial $P_t(x)=x^4-tx^3-6x^2+tx+1$, assuming that $t>0$, $t\neq 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in all fields in this family.

Para>In 1882, Lindemann wrote a paper in which he sketched a general result, special cases of which imply the transcendence of e and π. This general result was later proved rigorously by K. Weierstrass in 1885. Before we begin, we make some remarks pertaining to algebraic number theory. Let K be an algebraic number field.