''Basic Number Theory'' is an influential book by

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is du ...

, an exposition of

algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

and

class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

with particular emphasis on valuation-theoretic methods. Based in part on a course taught at

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

in 1961–62, it appeared as Volume 144 in Springer's ''Grundlehren der mathematischen Wissenschaften'' series. The approach handles all 'A-fields' or

global field In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global functio ...

s, meaning finite

algebraic extension In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, every element of is a root of a non-zero polynomial with coefficients in . A field extens ...

s of the field of

rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...

and of the field of

rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...

s of one variable with a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

of constants. The theory is developed in a uniform way, starting with topological fields, properties of

Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfr� ...

locally compact field In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space.. These kinds of fields were originally introduced in p-adic analysis since the fields \mathbb_p of p-adic numbers are locally compact ...

s, the main theorems of adelic and idelic number theory, and class field theory via the theory of

simple algebra In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a sim ...

s over local and global fields. The word `basic’ in the title is closer in meaning to `foundational’ rather than `elementary’, and is perhaps best interpreted as meaning that the material developed is foundational for the development of the theories of

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G o ...

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

s, and more advanced topics in algebraic number theory. The style is austere, with a narrow concentration on a logically coherent development of the theory required, and essentially no examples.

Mathematical context and purpose

In the foreword, the author explains that instead of the “futile and impossible task” of improving on Hecke's classical treatment of algebraic number theory, he “rather tried to draw the conclusions from the developments of the last thirty years, whereby locally compact groups, measure and integration have been seen to play an increasingly important role in classical number theory”. Weil goes on to explain a viewpoint that grew from work of Hensel, Hasse, Chevalley,

Artin Artin may refer to: * Artin (name), a surname and given name, including a list of people with the name ** Artin, a variant of Harutyun Harutyun ( and in Western Armenian Յարութիւն) also spelled Haroutioun, Harutiun and its variants Har ...

, Iwasawa,

Tate Tate is an institution that houses, in a network of four art galleries, the United Kingdom's national collection of British art, and international modern and contemporary art. It is not a government institution, but its main sponsor is the UK ...

, and Tamagawa in which the

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s may be seen as but one of infinitely many different completions of the rationals, with no logical reason to favour it over the various

p-adic In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infin ...

completions. In this setting, the adeles (or

valuation vector In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the Complete metric space, c ...

s) give a natural

locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...

ring in which all the valuations are brought together in a single coherent way in which they “cooperate for a common purpose”. Removing the real numbers from a pedestal and placing them alongside the p-adic numbers leads naturally – “it goes without saying” to the development of the theory of function fields over finite fields in a “fully simultaneous treatment with number-fields”. In a striking choice of wording for a foreword written in the United States in 1967, the author chooses to drive this particular viewpoint home by explaining that the two classes of

s “must be granted a fully simultaneous treatment ��instead of the segregated status, and at best the separate but equal facilities, which hitherto have been their lot. That, far from losing by such treatment, both races stand to gain by it, is one fact which will, I hope, clearly emerge from this book.” After

World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

, a series of developments in

diminished the significance of the

cyclic algebra In algebra, a cyclic division algebra is one of the basic examples of a division algebra over a field and plays a key role in the theory of central simple algebras. Definition Let ''A'' be a finite-dimensional central simple algebra over a field ...

s (and, more generally, the crossed product algebras) which are defined in terms of the number field in proofs of class field theory. Instead cohomological formalism became a more significant part of local and global class field theory, particularly in work of Hochschild and Nakayama, Weil,

, and

during the period 1950–1952. Alongside the desire to consider

algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

s alongside function fields over finite fields, the work of Chevalley is particularly emphasised. In order to derive the theorems of global class field theory from those of

local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite re ...

, Chevalley introduced what he called the élément idéal, later called idèle, at Hasse's suggestion. The idèle group of a

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

was first introduced by Chevalley in order to describe global class field theory for infinite extensions, but several years later he used it in a new way to derive global class field theory from local class field theory. Weil mentioned this (unpublished) work as a significant influence on some of the choices of treatment he uses.

Reception

The 1st edition was reviewed by George Whaples for

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

and Helmut Koch for Zentralblatt.Helmut Koch, (1st ed.), (2nd ed.) Later editions were reviewed by Fernando Q. Gouvêa for the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary edu ...

and by Koch for Zentralblatt; in his review of the second edition Koch makes the remark "

Shafarevich Igor Rostislavovich Shafarevich (; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. Outside mathematics, he wrote books and articles that criticised social ...

showed me the first edition in autumn 1967 in Moscow and said that this book will be from now on the book about class field theory". The coherence of the treatment, and some of its distinctive features, were highlighted by several reviewers, with Koch going on to say "This book is written in the spirit of the early forties and just this makes it a valuable source of information for everyone who is working about problems related to number and function fields."

Roughly speaking, the first half of the book is modern in its consistent use of adelic and id''è''lic methods and the simultaneous treatment of algebraic number fields and rational function fields over finite fields. The second half is arguably pre-modern in its development of

s and

without the language of

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

, and without the language of

Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated with a field extension ''L''/''K'' acts in a na ...

in particular. The author acknowledges this as a trade-off, explaining that “to develop such an approach systematically would have meant loading a great deal of unnecessary machinery on a ship which seemed well equipped for this particular voyage; instead of making it more seaworthy, it might have sunk it.” The treatment of class field theory uses analytic methods on both commutative fields and simple algebras. These methods show their power in giving the first unified proof that if K/k is a finite

normal extension In abstract algebra, a normal extension is an Algebraic extension, algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' that has a zero of a function, root in ''L'' splits into linear factors over ''L''. This is ...

of A-fields, then any

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...

of K over k is induced by the

Frobenius automorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class that includes finite fields. The endomorphism m ...

for infinitely many places of K. This approach also allows for a significantly simpler and more logical proof of algebraic statements, for example the result that a simple algebra over an A-field splits (globally) if and only if it splits everywhere locally. The systematic use of simple algebras also simplifies the treatment of

. For instance, it is more straightforward to prove that a simple algebra over a local field has an unramified

splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a polyn ...

than to prove the corresponding statement for 2-cohomology classes.

Chapter I

The book begins with Witt’s formulation of Wedderburn’s proof that a finite division ring is commutative ('

Wedderburn's little theorem In mathematics, Wedderburn's little theorem states that every finite division ring is a field; thus, every finite domain is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields. The Art ...

'). Properties of

are used to prove that `local fields’ (commutative fields locally compact under a non-discrete topology) are completions of A-fields. In particular – a concept developed later – they are precisely the fields whose local class field theory is needed for the global theory. The non-discrete non-commutative locally compact fields are then

division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...

s of finite dimension over a local field.

Chapter II

Finite-dimensional vector spaces over local fields and division algebras under the topology uniquely determined by the field's topology are studied, and lattices are defined topologically, an analogue of Minkowski's theorem is proved in this context, and the main theorems about

character group In mathematics, a character group is the group of representations of an abelian group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters that a ...

s of these vector spaces, which in the commutative one-dimensional case reduces to `self duality’ for local fields, are shown.

Chapter III

Tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...

s are used to study extensions of the places of an A-field to places of a finite

separable extension In field theory (mathematics), field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial (field theory), minimal polynomial of \alpha over is a separable po ...

of the field, with the more complicated inseparable case postponed to later.

Chapter IV

This chapter introduces the topological

adele ring In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the glob ...

and idèle group of an A-field, and proves the `main theorems’ as follows: * both the adele ring and the idèle group are locally compact; * the A-field, when embedded diagonally, is a discrete and co-compact subring of its adele ring; * the adele ring is self dual, meaning that it is topologically isomorphic to its

Pontryagin dual In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...

, with similar properties for finite-dimensional vector spaces and algebras over local fields. The chapter ends with a generalized unit theorem for A-fields, describing the units in valuation terms.

Chapter V

This chapter departs slightly from the simultaneous treatment of number fields and function fields. In the number field setting, lattices (that is,

fractional ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral do ...

s) are defined, and the Haar measure volume of a

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each ...

for a lattice is found. This is used to study the

discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...

of an extension.

Chapter VI

This chapter is focused on the function field case; the Riemann-Roch theorem is stated and proved in measure-theoretic language, with the

canonical class In mathematics, the canonical bundle of a non-singular variety, non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the nth exterior power of the cotangent bundle \Omega on V. Over the c ...

defined as the class of divisors of non-trivial characters of the

which are trivial on the embedded field.

Chapter VII

The

zeta Zeta (, ; uppercase Ζ, lowercase ζ; , , classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived from the Phoenician alphabet, Phoenician letter zay ...

and

L-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may gi ...

s (and similar analytic objects) for an A-field are expressed in terms of integrals over the idèle group. Decomposing these integrals into products over all valuations and using

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

s gives rise to meromorphic continuations and

functional equations In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

. This gives, for example,

analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...

of the Dedekind zeta-function to the whole plane, along with its functional equation. The treatment here goes back ultimately to a suggestion of

, and was developed in

Tate's thesis In number theory, Tate's thesis is the 1950 PhD thesis of completed under the supervision of Emil Artin at Princeton University. In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta functi ...

Chapter VIII

Formulas for local and global different and discriminants, ramification theory, and the formula for the

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

of an algebraic extension of a function field are developed.

Chapter IX

A brief treatment of simple algebras is given, including explicit rules for cyclic factor sets.

Chapters X and XI

The zeta-function of a simple algebra over an A-field is defined, and used to prove further results on the norm group and

groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...

maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...

s in a simple algebra over an A-field.

Chapter XII

The

reciprocity law In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an ir ...

over a local field in the context of a pairing of the

multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...

of a field and the

of the

absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...

of the

algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ...

of the field is proved. Ramification theory for

abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galoi ...

s is developed.

Chapter XIII

The global class field theory for A-fields is developed using the pairings of Chapter XII, replacing multiplicative groups of local fields with idèle class groups of A-fields. The pairing is constructed as a product over places of local Hasse invariants.

Third edition

Some references are added, some minor corrections made, some comments added, and five appendices are included, containing the following material: * A character version of the (local) transfer theorem and its extension to the global transfer theorem. * Šafarevič's theorem on the structure of

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

s of local fields using the theory of

Weil group In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field ''F'', its Weil group is generally denoted ''WF''. There also exists "finite leve ...

s. * Theorems of

and Sen on the Herbrand distribution. * Examples of L-functions with Grössencharacter.

Editions

References

{{reflist Mathematics books Algebraic number theory