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63-624: Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory . He was born in Philadelphia , United States , in a Jewish family of Lithuanian extraction. Mordell was educated at the University of Cambridge where he completed the Cambridge Mathematical Tripos as a student of St John's College, Cambridge , starting in 1906 after successfully passing

126-435: A {\displaystyle c/a} , presumably for actual use as a "table", for example, with a view to applications. It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy , for example, truly came into its own only later. It has been suggested instead that the table was a source of numerical examples for school problems. While evidence of Babylonian number theory

189-652: A 1 mod m 1 {\displaystyle n\equiv a_{1}{\bmod {m}}_{1}} , n ≡ a 2 mod m 2 {\displaystyle n\equiv a_{2}{\bmod {m}}_{2}} could be solved by a method he called kuṭṭaka , or pulveriser ; this is a procedure close to (a generalisation of) the Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations. Brahmagupta (628 AD) started

252-491: A Diophantine equations a polynomial equations to which rational or integer solutions are sought. While Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry , it seems to be the case that Indian mathematics is otherwise an indigenous tradition; in particular, there is no evidence that Euclid's Elements reached India before the 18th century. Āryabhaṭa (476–550 AD) showed that pairs of simultaneous congruences n ≡

315-557: A Fellow of St. John's , when elected to the Sadleirian Chair , and became Head of Department. He officially retired in 1953. It was at this time that he had his only formal research students, of whom J. W. S. Cassels was one. His idea of supervising research was said to involve the suggestion that a proof of the transcendence of the Euler–Mascheroni constant was probably worth a doctorate. His book Diophantine Equations (1969)

378-498: A disciple of Theodorus's; he worked on distinguishing different kinds of incommensurables , and was thus arguably a pioneer in the study of number systems . (Book X of Euclid's Elements is described by Pappus as being largely based on Theaetetus's work.) Euclid devoted part of his Elements to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic to it (Books VII to IX of Euclid's Elements ). In particular, he gave an algorithm for computing

441-455: A first foray towards both Évariste Galois 's work and algebraic number theory . Starting early in the nineteenth century, the following developments gradually took place: Algebraic number theory may be said to start with the study of reciprocity and cyclotomy , but truly came into its own with the development of abstract algebra and early ideal theory and valuation theory; see below. A conventional starting point for analytic number theory

504-529: A keen interest in mathematics, and distinguished clearly between arithmetic and calculation. (By arithmetic he meant, in part, theorising on number, rather than what arithmetic or number theory have come to mean.) It is through one of Plato's dialogues—namely, Theaetetus —that it is known that Theodorus had proven that 3 , 5 , … , 17 {\displaystyle {\sqrt {3}},{\sqrt {5}},\dots ,{\sqrt {17}}} are irrational. Theaetetus was, like Plato,

567-407: A naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by Kummer ) seems to have come from the study of higher reciprocity laws, that is, generalisations of quadratic reciprocity . Number fields are often studied as extensions of smaller number fields: a field L is said to be an extension of a field K if L contains K . (For example,

630-473: A non-elementary one. Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics. Analytic number theory may be defined Some subjects generally considered to be part of analytic number theory, for example, sieve theory , are better covered by

693-627: A treatise on squares in arithmetic progression by Fibonacci —who traveled and studied in north Africa and Constantinople—no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the late Renaissance , thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' Arithmetica . Pierre de Fermat (1607–1665) never published his writings; in particular, his work on number theory

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756-408: Is Dirichlet's theorem on arithmetic progressions (1837), whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable. The first use of analytic ideas in number theory actually goes back to Euler (1730s), who used formal power series and non-rigorous (or implicit) limiting arguments. The use of complex analysis in number theory comes later:

819-681: Is a totally real field of conductor m , then where φ is the Euler totient function and [ F  : Q {\displaystyle \mathbb {Q} } ] is the degree of F over Q {\displaystyle \mathbb {Q} } . Stickelberger's Theorem Let F be an abelian number field. Then, the Stickelberger ideal of F annihilates the class group of F . Note that θ ( F ) itself need not be an annihilator, but any multiple of it in Z {\displaystyle \mathbb {Z} } [ G F ] is. Explicitly,

882-485: Is a fixed rational number whose square root is not rational.) For that matter, the 11th-century chakravala method amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such. The grounds of the subject were set in the late nineteenth century, when ideal numbers , the theory of ideals and valuation theory were introduced; these are three complementary ways of dealing with

945-441: Is an algebraic number. Fields of algebraic numbers are also called algebraic number fields , or shortly number fields . Algebraic number theory studies algebraic number fields. Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study. It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as

1008-449: Is an ideal in the group ring Z {\displaystyle \mathbb {Z} } [ G m ] . They are defined as follows. Let ζ m denote a primitive m th root of unity . The isomorphism from ( Z {\displaystyle \mathbb {Z} } / m Z {\displaystyle \mathbb {Z} } ) to G m is given by sending a to σ a defined by the relation The Stickelberger element of level m

1071-456: Is based on lectures, and gives an idea of his discursive style. Mordell is said to have hated administrative duties. While visiting the University of Calgary , the elderly Mordell attended the Number Theory seminars and would frequently fall asleep during them. According to a story by number theorist Richard K. Guy , the department head at the time, after Mordell had fallen asleep, someone in

1134-589: Is commonly preferred as an adjective to number-theoretic . The earliest historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 ( Larsa, Mesopotamia , ca. 1800 BC) contains a list of " Pythagorean triples ", that is, integers ( a , b , c ) {\displaystyle (a,b,c)} such that a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} . The triples are too many and too large to have been obtained by brute force . The heading over

1197-433: Is contained almost entirely in letters to mathematicians and in private marginal notes. In his notes and letters, he scarcely wrote any proofs—he had no models in the area. Over his lifetime, Fermat made the following contributions to the field: The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur Goldbach , pointed him towards some of Fermat's work on

1260-482: Is defined as The Stickelberger ideal of level m , denoted I ( K m ) , is the set of integral multiples of θ ( K m ) which have integral coefficients, i.e. More generally, if F be any Abelian number field whose Galois group over Q {\displaystyle \mathbb {Q} } is denoted G F , then the Stickelberger element of F and the Stickelberger ideal of F can be defined. By

1323-464: Is irrational is credited to the early Pythagoreans (pre- Theodorus ). By revealing (in modern terms) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; its proof or its divulgation are sometimes credited to Hippasus , who was expelled or split from the Pythagorean sect. This forced a distinction between numbers (integers and

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1386-638: Is only survived by the Plimpton 322 tablet, some authors assert that Babylonian algebra was exceptionally well developed and included the foundations of modern elementary algebra . Late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt . In book nine of Euclid's Elements , propositions 21–34 are very probably influenced by Pythagorean teachings ; it

1449-436: Is then defined as The Stickelberger ideal of F , denoted I ( F ) , is defined as in the case of K m , i.e. In the special case where F = K m , the Stickelberger ideal I ( K m ) is generated by ( a − σ a ) θ ( K m ) as a varies over Z {\displaystyle \mathbb {Z} } / m Z {\displaystyle \mathbb {Z} } . This not true for general F . If F

1512-457: Is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions , which are generalizations of the Riemann zeta function , a key analytic object at

1575-465: Is to determine when a Diophantine equation has solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object. Stickelberger%27s theorem In mathematics , Stickelberger's theorem is a result of algebraic number theory , which gives some information about the Galois module structure of class groups of cyclotomic fields . A special case

1638-449: Is used by the general public to mean " elementary calculations "; it has also acquired other meanings in mathematical logic , as in Peano arithmetic , and computer science , as in floating-point arithmetic .) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical

1701-425: Is very simple material ("odd times even is even", "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it"), but it is all that is needed to prove that 2 {\displaystyle {\sqrt {2}}} is irrational . Pythagorean mystics gave great importance to the odd and the even. The discovery that 2 {\displaystyle {\sqrt {2}}}

1764-508: The Kronecker–Weber theorem there is an integer m such that F is contained in K m . Fix the least such m (this is the (finite part of the) conductor of F over Q {\displaystyle \mathbb {Q} } ). There is a natural group homomorphism G m → G F given by restriction, i.e. if σ ∈ G m , its image in G F is its restriction to F denoted res m σ . The Stickelberger element of F

1827-627: The University of Manchester in 1922 and Professor in 1923. There he developed a third area of interest within number theory, the geometry of numbers . His basic work on Mordell's theorem is from 1921 to 1922, as is the formulation of the Mordell conjecture . He was an Invited Speaker of the International Congress of Mathematicians (ICM) in 1928 in Bologna and in 1932 in Zürich and a Plenary Speaker of

1890-412: The multiplicative group of integers modulo m ( Z {\displaystyle \mathbb {Z} } / m Z {\displaystyle \mathbb {Z} } ) . The Stickelberger element ( of level m or of K m ) is an element in the group ring Q {\displaystyle \mathbb {Q} } [ G m ] and the Stickelberger ideal ( of level m or of K m )

1953-656: The ICM in 1936 in Oslo . He took British citizenship in 1929. In Manchester he also built up the department, offering posts to a number of outstanding mathematicians who had been forced from posts on the continent of Europe. He brought in Reinhold Baer , G. Billing, Paul Erdős , Chao Ko , Kurt Mahler , and Beniamino Segre . He also recruited J. A. Todd , Patrick du Val , Harold Davenport and Laurence Chisholm Young , and invited distinguished visitors. In 1945, he returned to Cambridge as

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2016-565: The audience asked "Isn't that Stickelberger's theorem ?" The speaker said "No it isn't." A few minutes later the person interrupted again and said "I'm positive that's Stickelberger's theorem!" The speaker again said no it wasn't. The lecture ended, and the applause woke up Mordell, and he looked up and pointed at the board, saying "There's old Stickelberger 's result!" Number theory Integers can be considered either in themselves or as solutions to equations ( Diophantine geometry ). Questions in number theory are often best understood through

2079-474: The basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied quadratic forms in full generality (as opposed to m X 2 + n Y 2 {\displaystyle mX^{2}+nY^{2}} )—defining their equivalence relation, showing how to put them in reduced form, etc. Adrien-Marie Legendre (1752–1833)

2142-625: The caliph Al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the Sindhind , which may or may not be Brahmagupta's Brāhmasphuṭasiddhānta ). Diophantus's main work, the Arithmetica , was translated into Arabic by Qusta ibn Luqa (820–912). Part of the treatise al-Fakhri (by al-Karajī , 953 – ca. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn al-Haytham knew what would later be called Wilson's theorem . Other than

2205-403: The complex numbers C are an extension of the reals R , and the reals R are an extension of the rationals Q .) Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions L of K such that the Galois group Gal( L / K ) of L over K is an abelian group —are relatively well understood. Their classification

2268-496: The discussion of quadratic forms in Disquisitiones arithmeticae can be restated in terms of ideals and norms in quadratic fields. (A quadratic field consists of all numbers of the form a + b d {\displaystyle a+b{\sqrt {d}}} , where a {\displaystyle a} and b {\displaystyle b} are rational numbers and d {\displaystyle d}

2331-394: The first column reads: "The takiltum of the diagonal which has been subtracted such that the width..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity which is implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and then reordered by c /

2394-505: The greatest common divisor of two numbers (the Euclidean algorithm ; Elements , Prop. VII.2) and the first known proof of the infinitude of primes ( Elements , Prop. IX.20). In 1773, Lessing published an epigram he had found in a manuscript during his work as a librarian; it claimed to be a letter sent by Archimedes to Eratosthenes . The epigram proposed what has become known as Archimedes's cattle problem ; its solution (absent from

2457-873: The lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationals and − 5 {\displaystyle {\sqrt {-5}}} , the number 6 {\displaystyle 6} can be factorised both as 6 = 2 ⋅ 3 {\displaystyle 6=2\cdot 3} and 6 = ( 1 + − 5 ) ( 1 − − 5 ) {\displaystyle 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})} ; all of 2 {\displaystyle 2} , 3 {\displaystyle 3} , 1 + − 5 {\displaystyle 1+{\sqrt {-5}}} and 1 − − 5 {\displaystyle 1-{\sqrt {-5}}} are irreducible, and thus, in

2520-405: The lines later developed fully by Gauss. In his old age, he was the first to prove Fermat's Last Theorem for n = 5 {\displaystyle n=5} (completing work by Peter Gustav Lejeune Dirichlet , and crediting both him and Sophie Germain ). In his Disquisitiones Arithmeticae (1798), Carl Friedrich Gauss (1777–1855) proved the law of quadratic reciprocity and developed

2583-505: The manuscript) requires solving an indeterminate quadratic equation (which reduces to what would later be misnamed Pell's equation ). As far as it is known, such equations were first successfully treated by the Indian school . It is not known whether Archimedes himself had a method of solution. Very little is known about Diophantus of Alexandria ; he probably lived in the third century AD, that is, about five hundred years after Euclid. Six out of

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2646-490: The mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period . In the case of number theory, this means, by and large, Plato and Euclid , respectively. While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition. Eusebius , PE X, chapter 4 mentions of Pythagoras : "In fact

2709-426: The most important tools of analytic number theory are the circle method , sieve methods and L-functions (or, rather, the study of their properties). The theory of modular forms (and, more generally, automorphic forms ) also occupies an increasingly central place in the toolbox of analytic number theory. One may ask analytic questions about algebraic numbers , and use analytic means to answer such questions; it

2772-694: The most interesting questions in each area remain open and are being actively worked on. The term elementary generally denotes a method that does not use complex analysis . For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg . The term is somewhat ambiguous: for example, proofs based on complex Tauberian theorems (for example, Wiener–Ikehara ) are often seen as quite enlightening but not elementary, in spite of using Fourier analysis , rather than complex analysis as such. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than

2835-581: The multiplicative property of Srinivasa Ramanujan 's tau-function . The proof was by means, in effect, of the Hecke operators , which had not yet been named after Erich Hecke ; it was, in retrospect, one of the major advances in modular form theory, beyond its status as an odd corner of the theory of special functions . In 1920, he took a teaching position in UMIST , becoming the Fielden Chair of Pure Mathematics at

2898-399: The rationals—the subjects of arithmetic), on the one hand, and lengths and proportions (which may be identified with real numbers, whether rational or not), on the other hand. The Pythagorean tradition spoke also of so-called polygonal or figurate numbers . While square numbers , cubic numbers , etc., are seen now as more natural than triangular numbers , pentagonal numbers , etc.,

2961-755: The roots of the subject. This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function. An algebraic number is any complex number that is a solution to some polynomial equation f ( x ) = 0 {\displaystyle f(x)=0} with rational coefficients; for example, every solution x {\displaystyle x} of x 5 + ( 11 / 2 ) x 3 − 7 x 2 + 9 = 0 {\displaystyle x^{5}+(11/2)x^{3}-7x^{2}+9=0} (say)

3024-520: The said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and Egypt, and all Persia, being instructed by the Magi and the priests: and in addition to these he is related to have studied under the Brahmans (these are Indian philosophers); and from some he gathered astrology, from others geometry, and arithmetic and music from others, and different things from different nations, and only from

3087-475: The scholarship examination. He graduated as third wrangler in 1909. After graduating Mordell began independent research into particular diophantine equations : the question of integer points on the cubic curve , and special case of what is now called a Thue equation , the Mordell equation He took an appointment at Birkbeck College, London in 1913. During World War I he was involved in war work, but also produced one of his major results, proving in 1917

3150-559: The second rather than the first definition: some of sieve theory, for instance, uses little analysis, yet it does belong to analytic number theory. The following are examples of problems in analytic number theory: the prime number theorem , the Goldbach conjecture (or the twin prime conjecture , or the Hardy–Littlewood conjectures ), the Waring problem and the Riemann hypothesis . Some of

3213-493: The study of analytical objects (for example, the Riemann zeta function ) that encode properties of the integers, primes or other number-theoretic objects in some fashion ( analytic number theory ). One may also study real numbers in relation to rational numbers; for example, as approximated by the latter ( Diophantine approximation ). The older term for number theory is arithmetic . By the early twentieth century, it had been superseded by number theory . (The word arithmetic

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3276-512: The study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period (17th to early 19th centuries). The Chinese remainder theorem appears as an exercise in Sunzi Suanjing (3rd, 4th or 5th century CE). (There is one important step glossed over in Sunzi's solution: it is the problem that was later solved by Āryabhaṭa 's Kuṭṭaka – see below .) The result

3339-399: The subject. This has been called the "rebirth" of modern number theory, after Fermat's relative lack of success in getting his contemporaries' attention for the subject. Euler's work on number theory includes the following: Joseph-Louis Lagrange (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations—for instance, the four-square theorem and

3402-452: The systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation , in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala , or "cyclic method") for solving Pell's equation

3465-518: The theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation ( congruences ) and devoted a section to computational matters, including primality tests. The last section of the Disquisitiones established a link between roots of unity and number theory: The theory of the division of the circle...which is treated in sec. 7 does not belong by itself to arithmetic, but its principles can only be drawn from higher arithmetic. In this way, Gauss arguably made

3528-540: The thirteen books of Diophantus's Arithmetica survive in the original Greek and four more survive in an Arabic translation. The Arithmetica is a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form f ( x , y ) = z 2 {\displaystyle f(x,y)=z^{2}} or f ( x , y , z ) = w 2 {\displaystyle f(x,y,z)=w^{2}} . Thus, nowadays,

3591-572: The wise men of Greece did he get nothing, wedded as they were to a poverty and dearth of wisdom: so on the contrary he himself became the author of instruction to the Greeks in the learning which he had procured from abroad." Aristotle claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans, and Cicero repeats this claim: Platonem ferunt didicisse Pythagorea omnia ("They say Plato learned all things Pythagorean"). Plato had

3654-423: The work of Bernhard Riemann (1859) on the zeta function is the canonical starting point; Jacobi's four-square theorem (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory ( modular forms ). The history of each subfield is briefly addressed in its own section below; see the main article of each subfield for fuller treatments. Many of

3717-452: Was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in Bhāskara II 's Bīja-gaṇita (twelfth century). Indian mathematics remained largely unknown in Europe until the late eighteenth century; Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke . In the early ninth century,

3780-493: Was first proven by Ernst Kummer ( 1847 ) while the general result is due to Ludwig Stickelberger ( 1890 ). Let K m denote the m th cyclotomic field , i.e. the extension of the rational numbers obtained by adjoining the m th roots of unity to Q {\displaystyle \mathbb {Q} } (where m ≥ 2 is an integer). It is a Galois extension of Q {\displaystyle \mathbb {Q} } with Galois group G m isomorphic to

3843-565: Was later generalized with a complete solution called Da-yan-shu ( 大衍術 ) in Qin Jiushao 's 1247 Mathematical Treatise in Nine Sections which was translated into English in early 19th century by British missionary Alexander Wylie . There is also some numerical mysticism in Chinese mathematics, but, unlike that of the Pythagoreans, it seems to have led nowhere. Aside from a few fragments,

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3906-402: Was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions . He gave a full treatment of the equation a x 2 + b y 2 + c z 2 = 0 {\displaystyle ax^{2}+by^{2}+cz^{2}=0} and worked on quadratic forms along

3969-524: Was the object of the programme of class field theory , which was initiated in the late 19th century (partly by Kronecker and Eisenstein ) and carried out largely in 1900–1950. An example of an active area of research in algebraic number theory is Iwasawa theory . The Langlands program , one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields. The central problem of Diophantine geometry

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