Algebraic number theory

Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory.

Algebraic number theory is a major branch of number theory that studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization, the behaviour of ideals, and field extensions. In this setting, the familiar features of the integers—such as unique factorization—need not hold. The virtue of the primary machinery employed—Galois theory, group cohomology, group representations, and L-functions—is that it allows one to deal with new phenomena and yet partially recover the behaviour of the usual integers.

History of algebraic number theory

Diophantus

The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively:

$A = x + y\$

$B = x^2 + y^2.\$

Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x² + y² = z² are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC). Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).

Diophantus's major work was the Arithmetica, of which only a portion has survived.

Fermat

Fermat's last theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century.

Gauss

One of the founding works of algebraic number theory, the Disquisitiones Arithmeticae (Latin: Arithmetical Investigations) is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own. Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways.

The Disquisitiones was the starting point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; we can now read them as containing the germs of the theories of L-functions and complex multiplication, in particular.

Dirichlet

In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms (later refined by his student Kronecker). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields. Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory.

He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem. He published important contributions to Fermat's last theorem, for which he proved the cases n = 5 and n = 14, and to the biquadratic reciprocity law. The Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.

Dedekind

Richard Dedekind's study of Lejeune Dirichlet's work was what led him to his later study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as Vorlesungen über Zahlentheorie ("Lectures on Number Theory") about which it has been written that:

"Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death." (Edwards 1983)

1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory. (The word "Ring", introduced later by Hilbert, does not appear in Dedekind's work.) Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether. Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem.

Hilbert

David Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results were mostly proved by 1930, after work by Teiji Takagi.

Artin

Emil Artin established the Artin reciprocity law in a series of papers (1924; 1927; 1930). This law is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.

Modern theory

Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct, branches of mathematics, elliptic curves and modular forms. The resulting modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.

It was initially dismissed as unlikely or highly speculative, and was taken more seriously when number theorist André Weil found evidence supporting it, but no proof; as a result the "astounding" conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of the Langlands programme, a list of important conjectures needing proof or disproof.

From 1993 to 1994, Andrew Wiles provided a proof of the modularity theorem for semistable elliptic curves, which, together with Ribet's theorem, provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the Modularity Theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians (meaning, impossible or virtually impossible to prove using current knowledge). Wiles first announced his proof in June 1993 in a version that was soon recognized as having a serious gap in a key point. The proof was corrected by Wiles, in part via collaboration with Richard Taylor, and the final, widely accepted, version was released in September 1994, and formally published in 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.

Basic notions

Unique factorization and the ideal class group

One of the first properties of Z that can fail in the ring of integers O of an algebraic number field K is that of the unique factorization of integers into prime numbers. The prime numbers in Z are generalized to irreducible elements in O, and though the unique factorization of elements of O into irreducible elements may hold in some cases (such as for the Gaussian integers Z[i]), it may also fail, as in the case of Z[√-5] where

$6=2\cdot3=(1+\sqrt{-5})\cdot(1-\sqrt{-5}).$

The ideal class group of O is a measure of how much unique factorization of elements fails; in particular, the ideal class group is trivial if, and only if, O is a unique factorization domain.

Factoring prime ideals in extensions

Unique factorization can be partially recovered for O in that it has the property of unique factorization of ideals into prime ideals (i.e. it is a Dedekind domain). This makes the study of the prime ideals in O particularly important. This is another area where things change from Z to O: the prime numbers, which generate prime ideals of Z (in fact, every single prime ideal of Z is of the form (p):=pZ for some prime number p,) may no longer generate prime ideals in O. For example, in the ring of Gaussian integers, the ideal 2Z[i] is no longer a prime ideal; in fact

$2\mathbf{Z}[i]=\left((1+i)\mathbf{Z}[i]\right)^2.$

On the other hand, the ideal 3Z[i] is a prime ideal. The complete answer for the Gaussian integers is obtained by using a theorem of Fermat's, with the result being that for an odd prime number p

$p\mathbf{Z}[i]\mbox{ is a prime ideal if }p\equiv 3 \,(\operatorname{mod}\, 4)$

$p\mathbf{Z}[i]\mbox{ is not a prime ideal if }p\equiv 1 \,(\operatorname{mod}\, 4).$

Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory. Class field theory accomplishes this goal when K is an abelian extension of Q (i.e. a Galois extension with abelian Galois group).

Primes and places

An important generalization of the notion of prime ideal in O is obtained by passing from the so-called ideal-theoretic approach to the so-called valuation-theoretic approach. The relation between the two approaches arises as follows. In addition to the usual absolute value function |·| : Q → R, there are absolute value functions |·|_p : Q → R defined for each prime number p in Z, called p-adic absolute values. Ostrowski's theorem states that these are all possible absolute value functions on Q (up to equivalence). This suggests that the usual absolute value could be considered as another prime. More generally, a prime of an algebraic number field K (also called a place) is an equivalence class of absolute values on K. The primes in K are of two sorts: $\mathfrak{p}$-adic absolute values like |·|_p, one for each prime ideal $\mathfrak{p}$ of O, and absolute values like |·| obtained by considering K as a subset of the complex numbers in various possible ways and using the absolute value |·| : C → R. A prime of the first kind is called a finite prime (or finite place) and one of the second kind is called an infinite prime (or infinite place). Thus, the set of primes of Q is generally denoted { 2, 3, 5, 7, ..., ∞ }, and the usual absolute value on Q is often denoted |·|_∞ in this context.

The set of infinite primes of K can be described explicitly in terms of the embeddings K → C (i.e. the non-zero ring homomorphisms from K to C). Specifically, the set of embeddings can be split up into two disjoint subsets, those whose image is contained in R, and the rest. To each embedding σ : K → R, there corresponds a unique prime of K coming from the absolute value obtained by composing σ with the usual absolute value on R; a prime arising in this fashion is called a real prime (or real place). To an embedding τ : K → C whose image is not contained in R, one can construct a distinct embedding τ, called the conjugate embedding, by composing τ with the complex conjugation map C → C. Given such a pair of embeddings τ and τ, there corresponds a unique prime of K again obtained by composing τ with the usual absolute value (composing τ instead gives the same absolute value function since |z| = |z| for any complex number z, where z denotes the complex conjugate of z). Such a prime is called a complex prime (or complex place). The description of the set of infinite primes is then as follows: each infinite prime corresponds either to a unique embedding σ : K → R, or a pair of conjugate embeddings τ, τ : K → C. The number of real (respectively, complex) primes is often denoted r₁ (respectively, r₂). Then, the total number of embeddings K → C is r₁+2r₂ (which, in fact, equals the degree of the extension K/Q).

Units

The fundamental theorem of arithmetic describes the multiplicative structure of Z. It states that every non-zero integer can be written (essentially) uniquely as a product of prime powers and ±1. The unique factorization of ideals in the ring O recovers part of this description, but fails to address the factor ±1. The integers 1 and -1 are the invertible elements (i.e. units) of Z. More generally, the invertible elements in O form a group under multiplication called the unit group of O, denoted O^×. This group can be much larger than the cyclic group of order 2 formed by the units of Z. Dirichlet's unit theorem describes the abstract structure of the unit group as an abelian group. A more precise statement giving the structure of O^× ⊗_Z Q as a Galois module for the Galois group of K/Q is also possible. The size of the unit group, and its lattice structure give important numerical information about O, as can be seen in the class number formula.

Local fields

Main article: Local field

Completing a number field K at a place w gives a complete field. If the valuation is archimedean, one gets R or C, if it is non-archimedean and lies over a prime p of the rationals, one gets a finite extension K_w / Q_p: a complete, discrete valued field with finite residue field. This process simplifies the arithmetic of the field and allows the local study of problems. For example, the Kronecker–Weber theorem can be deduced easily from the analogous local statement. The philosophy behind the study of local fields is largely motivated by geometric methods. In algebraic geometry, it is common to study varieties locally at a point by localizing to a maximal ideal. Global information can then be recovered by gluing together local data. This spirit is adopted in algebraic number theory. Given a prime in the ring of algebraic integers in a number field, it is desirable to study the field locally at that prime. Therefore, one localizes the ring of algebraic integers to that prime and then completes the fraction field much in the spirit of geometry.

Major results

Finiteness of the class group

One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field K is finite. The order of the class group is called the class number, and is often denoted by the letter h.

Dirichlet's unit theorem

Main article: Dirichlet's unit theorem

Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units O^× of the ring of integers O. Specifically, it states that O^× is isomorphic to G × Z^r, where G is the finite cyclic group consisting of all the roots of unity in O, and r = r₁ + r₂ − 1 (where r₁ (respectively, r₂) denotes the number of real embeddings (respectively, pairs of conjugate non-real embeddings) of K). In other words, O^× is a finitely generated abelian group of rank r₁ + r₂ − 1 whose torsion consists of the roots of unity in O.

Reciprocity laws

Main article: Reciprocity law

In terms of the Legendre symbol, the law of quadratic reciprocity for positive odd primes states

$\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}.$

A reciprocity law is a generalization of the law of quadratic reciprocity.

There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert symbols (a,b/p), taking values in roots of unity, is equal to 1. Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.

See also

Quadratic reciprocity

Cubic reciprocity

Quartic reciprocity

Artin reciprocity law

Class number formula

Main article: Class number formula

The class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.

Related areas

Algebraic number theory interacts with many other mathematical disciplines. It uses tools from homological algebra. Via the analogy of function fields vs. number fields, it relies on techniques and ideas from algebraic geometry. Moreover, the study of higher-dimensional schemes over Z instead of number rings is referred to as arithmetic geometry. Algebraic number theory is also used in the study of arithmetic hyperbolic 3-manifolds.

See also

• Langlands program
• Adele ring
• Tamagawa number
• Iwasawa theory
• Arithmetic algebraic geometry
• Class field theory
• Kummer theory
• Ideal class group

Further reading

Introductory texts

• Kenneth Ireland and Michael Rosen, "A Classical Introduction to Modern Number Theory, Second Edition", Springer-Verlag, 1990
• Ian Stewart and David O. Tall, "Algebraic Number Theory and Fermat's Last Theorem," A. K. Peters, 2002

Intermediate texts

• Daniel A. Marcus, "Number Fields"