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| [[Image:Disqvisitiones-800.jpg|thumb|Title page of the first edition of [[Disquisitiones Arithmeticae]], one of the founding works of modern algebraic number theory.]]
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| '''Algebraic number theory''' is a major branch of [[number theory]] which studies [[algebraic structure]]s related to [[algebraic integer]]s. This is generally accomplished by considering a [[Ring (mathematics)|ring]] of algebraic integers ''O'' in an [[algebraic number field]] ''K''/'''Q''', and studying their algebraic properties such as [[factorization]], the behaviour of [[Ideal (ring theory)|ideals]], and [[Field (mathematics)|field]] extensions. In this setting, the familiar features of the [[integer]]s—such as [[unique factorization]]—need not hold. The virtue of the primary machinery employed—[[Galois theory]], [[group cohomology]], [[group representation]]s, and [[L-function|''L''-functions]]—is that it allows one to deal with new phenomena and yet partially recover the behaviour of the usual integers.
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| ==History of algebraic number theory==
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| ===Diophantus===
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| The beginnings of algebraic number theory can be traced to [[Diophantine equation]]s,<ref>Stark, pp. 145–146.</ref> named for the 3rd-century [[Alexandria]]n 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:
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| :<math>A = x + y\ </math>
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| :<math>B = x^2 + y^2.\ </math>
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| Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation ''x''<sup>2</sup> + ''y''<sup>2</sup> = ''z''<sup>2</sup> are given by the [[Pythagorean triple]]s, originally solved by the Babylonians (c. 1800 BC).<ref>Aczel, pp. 14–15.</ref> Solutions to linear Diophantine equations, such as 26''x'' + 65''y'' = 13, may be found using the [[Euclidean algorithm]] (c. 5th century BC).<ref>Stark, pp. 44–47.</ref>
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| Diophantus's major work was the ''[[Arithmetica]]'', of which only a portion has survived.
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| ===Fermat===
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| [[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.
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| ===Gauss===
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| One of the founding works of algebraic number theory, the '''''Disquisitiones Arithmeticae''''' ([[Latin]]: ''Arithmetical Investigations'') is a textbook of [[number theory]] written in Latin<ref>[http://yalepress.yale.edu/yupbooks/book.asp?isbn=9780300094732 ''Disquisitiones Arithmeticae''] at Yalepress.yale.edu</ref> 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]], [[Joseph Louis Lagrange|Lagrange]] and [[Adrien-Marie Legendre|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.
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| The ''Disquisitiones'' was the starting point for the work of other nineteenth century [[Europe]]an 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-function]]s and [[complex multiplication]], in particular.
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| ===Dirichlet===
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| In a couple of papers in 1838 and 1839 [[Peter Gustav Lejeune Dirichlet]] proved the first [[class number formula]], for [[quadratic form]]s (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 field]]s.<ref name=Elstrodt> {{cite journal | last = Elstrodt | first = Jürgen | journal = Clay Mathematics Proceedings
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| | title = The Life and Work of Gustav Lejeune Dirichlet (1805–1859) | work = | publisher = | year = 2007
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| | url = http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | format = [[PDF]] | doi =
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| | accessdate = 2007-12-25}}</ref> Based on his research of the structure of the [[unit group]] of [[quadratic field]]s, he proved the [[Dirichlet unit theorem]], a fundamental result in [[algebraic number theory]].<ref name=Kanemitsu>{{cite book| last = Kanemitsu| first = Shigeru| coauthors = Chaohua Jia| title=Number theoretic methods: future trends | year=2002| publisher=Springer| location = | isbn= 978-1-4020-1080-4| pages= 271–274}}</ref>
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| 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 [[quartic reciprocity|biquadratic reciprocity law]].<ref name=Elstrodt/> 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.
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| ===Dedekind===
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| [[Richard Dedekind]]'s study of Lejeune Dirichlet's work was what led him to his later study of [[algebraic number field]]s and [[ideal (ring theory)|ideal]]s. 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:
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| <blockquote>"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)</blockquote>
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| 1879 and 1894 editions of the ''Vorlesungen'' included supplements introducing the notion of an ideal, fundamental to [[ring (algebra)|ring theory]]. (The word "Ring", introduced later by [[David Hilbert|Hilbert]], does not appear in Dedekind's work.) Dedekind defined an [[ring ideal|ideal]] as a subset of a set of numbers, composed of [[algebraic integer]]s 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 number]]s, devised as part of Kummer's 1843 attempt to prove [[Fermat's Last Theorem]].
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| ===Hilbert===
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| [[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 [[Waring's problem|problem formulated by Waring]] in 1770. As with [[#The finiteness theorem|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.<ref>Reid, Constance, 1996. ''Hilbert'', [[Springer Science and Business Media|Springer]], ISBN 0-387-94674-8.</ref> He then had little more to publish on the subject; but the emergence of [[Hilbert modular form]]s in the dissertation of a student means his name is further attached to a major area.
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| 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]].<ref>This work established Takagi as Japan's first mathematician of international stature.</ref>
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| ===Artin===
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| [[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]].<ref>[[Helmut Hasse]], ''History of Class Field Theory'', in ''Algebraic Number Theory'', edited by Cassels and Frölich, Academic Press, 1967, pp. 266–279</ref> The term "[[reciprocity law (mathematics)|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 [[Gotthold Eisenstein|Eisenstein]] and [[Ernst Kummer|Kummer]] to [[David Hilbert|Hilbert's]] product formula for the [[Hilbert symbol|norm symbol]]. Artin's result provided a partial solution to [[Hilbert's ninth problem]].
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| ===Modern theory===
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| Around 1955, Japanese mathematicians [[Goro Shimura]] and [[Yutaka Taniyama]] observed a possible link between two apparently completely distinct, branches of mathematics, [[elliptic curve]]s and [[modular form]]s. The resulting [[modularity theorem]] (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is [[modular elliptic curve|modular]], meaning that it can be associated with a unique [[modular form]].
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| 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"<ref name="Singh"> ''[[Fermat's Last Theorem (book)|Fermat's Last Theorem]]'', [[Simon Singh]], 1997, ISBN 1-85702-521-0></ref> 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.
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| From 1993 to 1994, [[Andrew Wiles]] provided a [[Mathematical proof|proof]] of the [[modularity theorem]] for [[semistable elliptic curve]]s, 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<ref name=nyt>{{cite news|last=Kolata|first=Gina|title=At Last, Shout of 'Eureka!' In Age-Old Math Mystery|url=http://www.nytimes.com/1993/06/24/us/at-last-shout-of-eureka-in-age-old-math-mystery.html|accessdate=21 January 2013|newspaper=The New York Times|date=24 June 1993}}</ref> 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 (mathematician)|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 (mathematics)|category]] of [[scheme (mathematics)|schemes]] and [[Iwasawa theory]], and other 20th-century techniques not available to Fermat.
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| ==Basic notions==
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| ===Unique factorization and the ideal class group===
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| 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 number]]s. The prime numbers in '''Z''' are generalized to [[irreducible element]]s 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'''[√{{Overline|-5}}] where
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| :<math> 6=2\cdot3=(1+\sqrt{-5})\cdot(1-\sqrt{-5}).</math>
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| 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 group|trivial]] if, and only if, ''O'' is a [[unique factorization domain]].
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| ===Factoring prime ideals in extensions===
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| Unique factorization can be partially recovered for ''O'' in that it has the property of unique factorization of ''ideals'' into [[prime ideal]]s (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 [[Principal ideal|generate]] prime ideals of '''Z''' (in fact, every single prime ideal of '''Z''' is of the form (''p''):=''p'''''Z''' for some prime number ''p'',) may no longer generate prime ideals in ''O''. For example, in the ring of Gaussian integers, the ideal 2'''Z'''[i] is no longer a prime ideal; in fact
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| :<math>2\mathbf{Z}[i]=\left((1+i)\mathbf{Z}[i]\right)^2.</math>
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| On the other hand, the ideal 3'''Z'''[i] is a prime ideal. The complete answer for the Gaussian integers is obtained by using a [[Fermat's theorem on sums of two squares|theorem of Fermat's]], with the result being that for an odd prime number ''p''
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| :<math>p\mathbf{Z}[i]\mbox{ is a prime ideal if }p\equiv 3 \,(\operatorname{mod}\, 4)</math>
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| :<math>p\mathbf{Z}[i]\mbox{ is not a prime ideal if }p\equiv 1 \,(\operatorname{mod}\, 4).</math>
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| 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 group|abelian]] [[Galois group]]).
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| ===Primes and places===
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| 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 [[Absolute value|usual absolute value]] function |·| : '''Q''' → '''R''', there are [[Absolute value (algebra)|absolute value]] functions |·|<sub>p</sub> : '''Q''' → '''R''' defined for each prime number ''p'' in '''Z''', called [[p-adic absolute value]]s. [[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: <math>\mathfrak{p}</math>-adic absolute values like |·|<sub>p</sub>, one for each prime ideal <math>\mathfrak{p}</math> of ''O'', and absolute values like |·| obtained by considering ''K'' as a subset of the [[complex number]]s 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 |·|<sub>∞</sub> in this context.
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| The set of infinite primes of ''K'' can be described explicitly in terms of the embeddings ''K'' → '''C''' (i.e. the non-zero [[ring homomorphism]]s from ''K'' to '''C'''). Specifically, the set of embeddings can be split up into two disjoint subsets, those whose [[Image (mathematics)|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 {{Overline|τ}}, called the ''conjugate embedding'', by composing τ with the [[complex conjugation]] map '''C''' → '''C'''. Given such a pair of embeddings τ and {{Overline|τ}}, there corresponds a unique prime of ''K'' again obtained by composing τ with the usual absolute value (composing {{Overline|τ}} instead gives the same absolute value function since |''z''| = |{{Overline|''z''}}| for any complex number ''z'', where {{Overline|''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, {{Overline|τ}} : ''K'' → '''C'''. The number of real (respectively, complex) primes is often denoted ''r''<sub>1</sub> (respectively, ''r''<sub>2</sub>). Then, the total number of embeddings ''K'' → '''C''' is ''r''<sub>1</sub>+2''r''<sub>2</sub> (which, in fact, equals the degree of the extension ''K''/'''Q''').
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| ===Units===
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| 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 power]]s 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. [[Unit (ring theory)|units]]) of '''Z'''. More generally, the invertible elements in ''O'' form a group under multiplication called the '''[[Unit (ring theory)#Group of units|unit group]]''' of ''O'', denoted ''O''<sup>×</sup>. 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''<sup>×</sup> ⊗<sub>'''Z'''</sub> '''Q''' as a [[Galois module]] for the Galois group of ''K''/'''Q''' is also possible.<ref>See proposition VIII.8.6.11 of {{harvnb|Neukirch|Schmidt|Wingberg|2000}}</ref> 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]].
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| ===Local fields===
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| {{main|Local field}}
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| [[Completion (metric space)|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''<sub>w</sub> / '''Q'''<sub>p</sub>: 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.
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| ==Major results==
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| ===Finiteness of the class group===
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| 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 (number theory)|class number]], and is often denoted by the letter ''h''.
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| ===Dirichlet's unit theorem===
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| {{Main|Dirichlet's unit theorem}}
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| Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units ''O''<sup>×</sup> of the ring of integers ''O''. Specifically, it states that ''O''<sup>×</sup> is isomorphic to ''G'' × '''Z'''<sup>''r''</sup>, where ''G'' is the finite cyclic group consisting of all the roots of unity in ''O'', and ''r'' = ''r''<sub>1</sub> + ''r''<sub>2</sub> − 1 (where ''r''<sub>1</sub> (respectively, ''r''<sub>2</sub>) denotes the number of real embeddings (respectively, pairs of conjugate non-real embeddings) of ''K''). In other words, ''O''<sup>×</sup> is a [[finitely generated abelian group]] of [[Rank of an abelian group|rank]] ''r''<sub>1</sub> + ''r''<sub>2</sub> − 1 whose torsion consists of the roots of unity in ''O''.
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| ===Reciprocity laws===
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| {{Main|Reciprocity law}}
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| In terms of the [[Legendre symbol]], the law of quadratic reciprocity for positive odd primes states
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| :<math> \left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}.</math>
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| A '''reciprocity law''' is a generalization of the [[law of quadratic reciprocity]].
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| 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 [[Legendre symbol|quadratic reciprocity symbol]], that describes when a [[prime number]] is an ''n''th power residue [[modular arithmetic|modulo]] another prime, and gave a relation between (''p''/''q'') and (''q''/''p''). [[David Hilbert|Hilbert]] reformulated the reciprocity laws as saying that a product over ''p'' of [[Hilbert symbol]]s (''a'',''b''/''p''), taking values in roots of unity, is equal to 1. [[Emil Artin|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.
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| See also
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| :[[Quadratic reciprocity]]
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| :[[Cubic reciprocity]]
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| :[[Quartic reciprocity]]
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| :[[Artin reciprocity law]]
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| ===Class number formula===
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| {{Main|Class number formula}}
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| The '''class number formula''' relates many important invariants of a [[number field]] to a special value of its [[Dedekind zeta function]].
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| ==Related areas==
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| 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-manifold]]s.
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| ==Notes==
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| <references/>
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| == References ==
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| ===Introductory texts===
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| * Kenneth Ireland and Michael Rosen, "A Classical Introduction to Modern Number Theory, Second Edition", Springer-Verlag, 1990
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| * [[Ian Stewart (mathematician)|Ian Stewart]] and [[David O. Tall]], "Algebraic Number Theory and Fermat's Last Theorem," A. K. Peters, 2002
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| ===Intermediate texts===
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| * Daniel A. Marcus, "Number Fields"
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| ===Graduate level accounts===
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| *{{Citation
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| | editor-last=Cassels
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| | editor-first=J. W. S.
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| | editor-link=J. W. S. Cassels
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| | editor2-last=Fröhlich
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| | editor2-first=Albrecht
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| | editor2-link=Albrecht Fröhlich
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| | title=Algebraic number theory
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| | year=1967
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| | place=London
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| | publisher=Academic Press
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| | mr=0215665
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| }}
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| *{{Citation
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| | last=Fröhlich
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| | first=Albrecht
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| | author-link=Albrecht Fröhlich
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| | last2=Taylor
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| | first2=Martin J.
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| | author2-link=Martin J. Taylor
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| | title=Algebraic number theory
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| | publisher=[[Cambridge University Press]]
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| | year=1993
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| | series=Cambridge Studies in Advanced Mathematics
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| | volume=27
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| | isbn=0-521-43834-9
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| | mr=1215934
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| }}
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| *{{Citation
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| | last=Lang
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| | first=Serge
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| | author-link=Serge Lang
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| | title=Algebraic number theory
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| | edition=2
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| | publisher=[[Springer-Verlag]]
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| | year=1994
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| | series=[[Graduate Texts in Mathematics]]
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| | volume=110
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| | place=New York
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| | isbn=978-0-387-94225-4
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| | mr=1282723
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| }}
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| *{{Neukirch ANT}}
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| ===Specific references===
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| * {{Neukirch et al. CNF|edition=1}}
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| ===External links===
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| * {{springer|title=Algebraic number theory|id=p/a011600}}
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| ==See also==
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| *[[Langlands program]]
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| *[[Adele ring]]
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| *[[Tamagawa number]]
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| *[[Iwasawa theory]]
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| *[[Arithmetic algebraic geometry]]
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| *[[Class field theory]]
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| {{Number theory-footer}}
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| [[Category:Algebraic number theory| ]]
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