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Algebraic number theory is a branch ofnumber theory that uses the techniques ofabstract algebra to study theintegers,rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such asalgebraic number fields and theirrings of integers,finite fields, andfunction fields. These properties, such as whether aring admitsunique factorization, the behavior ofideals, and theGalois groups offields, can resolve questions of primary importance in number theory, like the existence of solutions toDiophantine equations.
The beginnings of algebraic number theory can be traced to Diophantine equations,[1] named after the 3rd-centuryAlexandrian 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 integersx andy such that their sum, and the sum of their squares, equal two given numbersA andB, respectively:
Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation
x2 +y2 =z2 are given by thePythagorean triples, originally solved by the Babylonians (c. 1800 BC).[2] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using theEuclidean algorithm (c. 5th century BC).[3]
Diophantus's major work was theArithmetica, of which only a portion has survived.
Fermat's Last Theorem was firstconjectured byPierre de Fermat in 1637, famously in the margin of a copy ofArithmetica 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 themodularity theorem in the 20th century.
One of the founding works of algebraic number theory, theDisquisitiones Arithmeticae (Latin:Arithmetical Investigations) is a textbook of number theory written in Latin[4] byCarl 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 andLegendre and adds important new results of his own. Before theDisquisitiones 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.
TheDisquisitiones was the starting point for the work of other nineteenth centuryEuropean mathematicians includingErnst Kummer,Peter Gustav Lejeune Dirichlet andRichard 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 ofL-functions andcomplex multiplication, in particular.
In a couple of papers in 1838 and 1839Peter Gustav Lejeune Dirichlet proved the firstclass number formula, forquadratic forms (later refined by his studentLeopold Kronecker). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more generalnumber fields.[5] Based on his research of the structure of theunit group ofquadratic fields, he proved theDirichlet unit theorem, a fundamental result in algebraic number theory.[6]
He first used thepigeonhole principle, a basic counting argument, in the proof of a theorem indiophantine approximation, later named after himDirichlet's approximation theorem. He published important contributions to Fermat's last theorem, for which he proved the casesn = 5 andn = 14, and to thebiquadratic reciprocity law.[5] TheDirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.
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 asVorlesungen 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 theVorlesungen included supplements introducing the notion of an ideal, fundamental toring theory. (The word "Ring", introduced later byHilbert, does not appear in Dedekind's work.) Dedekind defined an ideal as a subset of a set of numbers, composed ofalgebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, ofEmmy Noether. Ideals generalize Ernst Eduard Kummer'sideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem.
David Hilbert unified the field of algebraic number theory with his 1897 treatiseZahlbericht (literally "report on numbers"). He also resolved a significant number-theoryproblem formulated by Waring in 1770. As withthe 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.[7] He then had little more to publish on the subject; but the emergence ofHilbert modular forms in the dissertation of a student means his name is further attached to a major area.
He made a series of conjectures onclass field theory. The concepts were highly influential, and his own contribution lives on in the names of theHilbert class field and of theHilbert symbol oflocal class field theory. Results were mostly proved by 1930, after work byTeiji Takagi.[8]
Emil Artin established theArtin 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.[9] The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from thequadratic reciprocity law and the reciprocity laws ofEisenstein and Kummer to Hilbert's product formula for thenorm symbol. Artin's result provided a partial solution toHilbert's ninth problem.
Around 1955, Japanese mathematiciansGoro Shimura andYutaka Taniyama observed a possible link between two apparently completely distinct, branches of mathematics,elliptic curves andmodular forms. The resultingmodularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve ismodular, meaning that it can be associated with a uniquemodular form.
It was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theoristAndré Weil found evidence supporting it, yet no proof; as a result the "astounding"[10] conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of theLanglands program, a list of important conjectures needing proof or disproof.
From 1993 to 1994,Andrew Wiles provided a proof of themodularity theorem forsemistable elliptic curves, which, together withRibet's theorem, provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting-edge developments. Wiles first announced his proof in June 1993[11] in a version that was soon recognized as having a serious gap at a key point. The proof was corrected by Wiles, partly in collaboration withRichard Taylor, and the final, widely accepted version was released in September 1994, and formally published in 1995. The proof uses many techniques fromalgebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as thecategory ofschemes andIwasawa theory, and other 20th-century techniques not available to Fermat.
An important property of the ring of integers is that it satisfies thefundamental theorem of arithmetic, that every (positive) integer has a factorization into a product ofprime numbers, and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integersO of an algebraic number fieldK.
Aprime element is an elementp ofO such that ifp divides a productab, then it divides one of the factorsa orb. This property is closely related to primality in the integers, because any positive integer satisfying this property is either1 or a prime number. However, it is strictly weaker. For example,−2 is not a prime number because it is negative, but it is a prime element. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as
In general, ifu is aunit, meaning a number with a multiplicative inverse inO, and ifp is a prime element, thenup is also a prime element. Numbers such asp andup are said to beassociate. In the integers, the primesp and−p are associate, but only one of these is positive. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements. WhenK is not the rational numbers, however, there is no analog of positivity. For example, in theGaussian integersZ[i],[12] the numbers1 + 2i and−2 +i are associate because the latter is the product of the former byi, but there is no way to single out one as being more canonical than the other. This leads to equations such as
which prove that inZ[i], it is not true that factorizations are unique up to the order of the factors. For this reason, one adopts the definition of unique factorization used inunique factorization domains (UFDs). In a UFD, the prime elements occurring in a factorization are only expected to be unique up to units and their ordering.
However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization. There is an algebraic obstruction called the ideal class group. When the ideal class group is trivial, the ring is a UFD. When it is not, there is a distinction between a prime element and anirreducible element. Anirreducible elementx is an element such that ifx =yz, then eithery orz is a unit. These are the elements that cannot be factored any further. Every element inO admits a factorization into irreducible elements, but it may admit more than one. This is because, while all prime elements are irreducible, some irreducible elements may not be prime. For example, consider the ringZ[√-5].[13] In this ring, the numbers3,2 + √-5 and2 - √-5 are irreducible. This means that the number9 has two factorizations into irreducible elements,
This equation shows that3 divides the product(2 + √-5)(2 - √-5) = 9. If3 were a prime element, then it would divide2 + √-5 or2 - √-5, but it does not, because all elements divisible by3 are of the form3a + 3b√-5. Similarly,2 + √-5 and2 - √-5 divide the product32, but neither of these elements divides3 itself, so neither of them are prime. As there is no sense in which the elements3,2 + √-5 and2 - √-5 can be made equivalent, unique factorization fails inZ[√-5]. Unlike the situation with units, where uniqueness could be repaired by weakening the definition, overcoming this failure requires a new perspective.
IfI is an ideal inO, then there is always a factorization
where each is aprime ideal, and where this expression is unique up to the order of the factors. In particular, this is true ifI is the principal ideal generated by a single element. This is the strongest sense in which the ring of integers of a general number field admits unique factorization. In the language of ring theory, it says that rings of integers areDedekind domains.
WhenO is a UFD, every prime ideal is generated by a prime element. Otherwise, there are prime ideals which are not generated by prime elements. InZ[√-5], for instance, the ideal(2, 1 + √-5) is a prime ideal which cannot be generated by a single element.
Historically, the idea of factoring ideals into prime ideals was preceded by Ernst Kummer's introduction of ideal numbers. These are numbers lying in an extension fieldE ofK. This extension field is now known as the Hilbert class field. By theprincipal ideal theorem, every prime ideal ofO generates a principal ideal of the ring of integers ofE. A generator of this principal ideal is called an ideal number. Kummer used these as a substitute for the failure of unique factorization incyclotomic fields. These eventually led Richard Dedekind to introduce a forerunner of ideals and to prove unique factorization of ideals.
An ideal which is prime in the ring of integers in one number field may fail to be prime when extended to a larger number field. Consider, for example, the prime numbers. The corresponding idealspZ are prime ideals of the ringZ. However, when this ideal is extended to the Gaussian integers to obtainpZ[i], it may or may not be prime. For example, the factorization2 = (1 +i)(1 −i) implies that
note that because1 +i = (1 −i) ⋅i, the ideals generated by1 +i and1 −i are the same. A complete answer to the question of which ideals remain prime in the Gaussian integers is provided byFermat's theorem on sums of two squares. It implies that for an odd prime numberp,pZ[i] is a prime ideal ifp ≡ 3 (mod 4) and is not a prime ideal ifp ≡ 1 (mod 4). This, together with the observation that the ideal(1 +i)Z[i] is prime, provides a complete description of the prime ideals in the Gaussian integers. Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory. Class field theory accomplishes this goal whenK is anabelian extension ofQ (that is, aGalois extension withabelian Galois group).
Unique factorization fails if and only if there are prime ideals that fail to be principal. The object which measures the failure of prime ideals to be principal is called the ideal class group. Defining the ideal class group requires enlarging the set of ideals in a ring of algebraic integers so that they admit agroup structure. This is done by generalizing ideals tofractional ideals. A fractional ideal is an additive subgroupJ ofK which is closed under multiplication by elements ofO, meaning thatxJ ⊆J ifx ∈O. All ideals ofO are also fractional ideals. IfI andJ are fractional ideals, then the setIJ of all products of an element inI and an element inJ is also a fractional ideal. This operation makes the set of non-zero fractional ideals into a group. The group identity is the ideal(1) =O, and the inverse ofJ is a (generalized)ideal quotient:
The principal fractional ideals, meaning the ones of the formOx wherex ∈K×, form a subgroup of the group of all non-zero fractional ideals. The quotient of the group of non-zero fractional ideals by this subgroup is the ideal class group. Two fractional idealsI andJ represent the same element of the ideal class group if and only if there exists an elementx ∈K such thatxI =J. Therefore, the ideal class group makes two fractional ideals equivalent if one is as close to being principal as the other is. The ideal class group is generally denotedClK,ClO, orPicO (with the last notation identifying it with thePicard group in algebraic geometry).
The number of elements in the class group is called theclass number ofK. The class number ofQ(√-5) is 2. This means that there are only two ideal classes, the class of principal fractional ideals, and the class of a non-principal fractional ideal such as(2, 1 + √-5).
The ideal class group has another description in terms ofdivisors. These are formal objects which represent possible factorizations of numbers. The divisor groupDivK is defined to be thefree abelian group generated by the prime ideals ofO. There is agroup homomorphism fromK×, the non-zero elements ofK up to multiplication, toDivK. Suppose thatx ∈K satisfies
Thendivx is defined to be the divisor
Thekernel ofdiv is the group of units inO, while thecokernel is the ideal class group. In the language ofhomological algebra, this says that there is anexact sequence of abelian groups (written multiplicatively),
Some number fields, such asQ(√2), can be specified as subfields of the real numbers. Others, such asQ(√−1), cannot. Abstractly, such a specification corresponds to a field homomorphismK →R orK →C. These are calledreal embeddings andcomplex embeddings, respectively.
A real quadratic fieldQ(√a), witha ∈Q,a > 0, anda not aperfect square, is so-called because it admits two real embeddings but no complex embeddings. These are the field homomorphisms which send√a to√a and to−√a, respectively. Dually, an imaginary quadratic fieldQ(√−a) admits no real embeddings but admits a conjugate pair of complex embeddings. One of these embeddings sends√−a to√−a, while the other sends it to itscomplex conjugate,−√−a.
Conventionally, the number of real embeddings ofK is denotedr1, while the number of conjugate pairs of complex embeddings is denotedr2. Thesignature ofK is the pair(r1,r2). It is a theorem thatr1 + 2r2 =d, whered is the degree ofK.
Considering all embeddings at once determines a function, or equivalentlyThis is called theMinkowski embedding.
The subspace of the codomain fixed by complex conjugation is a real vector space of dimensiond calledMinkowski space. Because the Minkowski embedding is defined by field homomorphisms, multiplication of elements ofK by an elementx ∈K corresponds to multiplication by adiagonal matrix in the Minkowski embedding. Thedot product on Minkowski space corresponds to the trace form.
The image ofO under the Minkowski embedding is ad-dimensionallattice. IfB is a basis for this lattice, thendetBTB is thediscriminant ofO. The discriminant is denotedΔ orD. The covolume of the image ofO is.
Real and complex embeddings can be put on the same footing as prime ideals by adopting a perspective based onvaluations. Consider, for example, the integers. In addition to the usualabsolute value function |·| :Q →R, there arep-adic absolute value functions |·|p :Q →R, defined for each prime numberp, which measure divisibility byp.Ostrowski's theorem states that these are all possible absolute value functions onQ (up to equivalence). Therefore, absolute values are a common language to describe both the real embedding ofQ and the prime numbers.
Aplace of an algebraic number field is anequivalence class ofabsolute value functions onK. There are two types of places. There is a-adic absolute value for each prime ideal ofO, and, like thep-adic absolute values, it measures divisibility. These are calledfinite places. The other type of place is specified using a real or complex embedding ofK and the standard absolute value function onR orC. These areinfinite places. Because absolute values are unable to distinguish between a complex embedding and its conjugate, a complex embedding and its conjugate determine the same place. Therefore, there arer1 real places andr2 complex places. Because places encompass the primes, places are sometimes referred to asprimes. When this is done, finite places are calledfinite primes and infinite places are calledinfinite primes. Ifv is a valuation corresponding to an absolute value, then one frequently writes to mean thatv is an infinite place and to mean that it is a finite place.
Considering all the places of the field together produces theadele ring of the number field. The adele ring allows one to simultaneously track all the data available using absolute values. This produces significant advantages in situations where the behavior at one place can affect the behavior at other places, as in theArtin reciprocity law.
There is a geometric analogy for places at infinity which holds on the function fields of curves. For example, let and be asmooth,projective,algebraic curve. Thefunction field has many absolute values, or places, and each corresponds to a point on the curve. If is the projective completion of an affine curve then the points in correspond to the places at infinity. Then, the completion of at one of these points gives an analogue of the-adics.
For example, if then its function field is isomorphic to where is an indeterminant and the field is the field of fractions of polynomials in. Then, a place at a point measures the order of vanishing or the order of a pole of a fraction of polynomials at the point. For example, if, so on the affine chart this corresponds to the point, the valuation measures theorder of vanishing of minus the order of vanishing of at. The function field of the completion at the place is then which is the field ofpower series in the variable, so an element is of the form
for some. For the place at infinity, this corresponds to the function field which are power series of the form
The integers have only two units,1 and−1. Other rings of integers may admit more units. The Gaussian integers have four units, the previous two as well as±i. TheEisenstein integersZ[exp(2πi / 3)] have six units. The integers in real quadratic number fields have infinitely many units. For example, inZ[√3], every power of2 + √3 is a unit, and all these powers are distinct.
In general, the group of units ofO, denotedO×, is a finitely generated abelian group. Thefundamental theorem of finitely generated abelian groups therefore implies that it is a direct sum of a torsion part and a free part. Reinterpreting this in the context of a number field, the torsion part consists of theroots of unity that lie inO. This group is cyclic. The free part is described byDirichlet's unit theorem. This theorem says that rank of the free part isr1 +r2 − 1. Thus, for example, the only fields for which the rank of the free part is zero areQ and the imaginary quadratic fields. A more precise statement giving the structure ofO× ⊗ZQ as aGalois module for the Galois group ofK/Q is also possible.[14]
The free part of the unit group can be studied using the infinite places ofK. Consider the function
wherev varies over the infinite places ofK and |·|v is the absolute value associated withv. The functionL is a homomorphism fromK× to a real vector space. It can be shown that the image ofO× is a lattice that spans the hyperplane defined by The covolume of this lattice is theregulator of the number field. One of the simplifications made possible by working with the adele ring is that there is a single object, theidele class group, that describes both the quotient by this lattice and the ideal class group.
TheDedekind zeta function of a number field, analogous to theRiemann zeta function, is an analytic object which describes the behavior of prime ideals inK. WhenK is an abelian extension ofQ, Dedekind zeta functions are products ofDirichlet L-functions, with there being one factor for eachDirichlet character. The trivial character corresponds to the Riemann zeta function. WhenK is aGalois extension, the Dedekind zeta function is theArtin L-function of theregular representation of the Galois group ofK, and it has a factorization in terms of irreducibleArtin representations of the Galois group.
The zeta function is related to the other invariants described above by theclass number formula.
Completing a number fieldK at a placew gives acomplete field. If the valuation is Archimedean, one obtainsR orC, if it is non-Archimedean and lies over a primep of the rationals, one obtains a finite extension 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, theKronecker–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.
One of the classical results in algebraic number theory is that the ideal class group of an algebraic number fieldK is finite. This is a consequence ofMinkowski's theorem since there are only finitely manyIntegral ideals with norm less than a fixed positive integer[15]page 78. The order of the class group is called theclass number, and is often denoted by the letterh.
Dirichlet's unit theorem provides a description of the structure of the multiplicative group of unitsO× of the ring of integersO. Specifically, it states thatO× is isomorphic toG ×Zr, whereG is the finitecyclic group consisting of all the roots of unity inO, andr =r1 + r2 − 1 (wherer1 (respectively,r2) denotes the number of real embeddings (respectively, pairs of conjugate non-real embeddings) ofK). In other words,O× is afinitely generated abelian group ofrankr1 + r2 − 1 whose torsion consists of the roots of unity inO.
In terms of theLegendre symbol, the law of quadratic reciprocity for positive odd primes states
Areciprocity law is a generalization of thelaw 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 apower residue symbol (p/q) generalizing thequadratic reciprocity symbol, that describes when aprime number is annth power residuemodulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product overp of Hilbert symbols (a,b/p), taking values in roots of unity, is equal to 1.Artin's reformulatedreciprocity law states 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.
Theclass number formula relates many important invariants of anumber field to a special value of its Dedekind zeta function.
Algebraic number theory interacts with many other mathematical disciplines. It uses tools fromhomological 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 overZ instead of number rings is referred to asarithmetic geometry. Algebraic number theory is also used in the study ofarithmetic hyperbolic 3-manifolds.
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