Algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, function fields; these properties, such as whether a ring admits unique factorization, the behavior of ideals, the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, 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, 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 solved by the Babylonians. Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm. Diophantus' major work was the Arithmetica. 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, 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. One of the founding works of algebraic number theory, the Disquisitiones Arithmeticae 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 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, 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 cryptic to his contemporaries. In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms; 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 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, 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 contributions by other researchers. Richard Dedekind's study of Lejeune Dirichlet's work was what led him to his study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as Vorlesungen über Zahlentheorie about which it has been written that: "Although the book is assuredly based on Dirichlet's lectures, although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was written by Dedekind, for the most part after Dirichlet's death." 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory.
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 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. David Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht, he 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 had little more to publish on the subject.
Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, algebraic number theory. He was awarded the Fields Medal in 1954 and the inaugural Abel Prize in 2003. Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and from 1945 to 1948 at the École Normale Supérieure in Paris, he was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994, his wife, Professor Josiane Heulot-Serre, was a chemist. Their daughter is the former French diplomat and writer Claudine Monteil; the French mathematician Denis Serre is his nephew. He practices skiing, table tennis, rock climbing. From a young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and commutative algebra and algebraic geometry, where he introduced sheaf theory and homological algebra techniques.
Serre's thesis concerned the Leray–Serre spectral sequence associated to a fibration. Together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, made the point that the award was for the first time awarded to a non-analyst. Serre subsequently changed his research focus. In the 1950s and 1960s, a fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents, on coherent cohomology, Géometrie Algébrique et Géométrie Analytique. At an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures; the problem was that the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology with integer coefficients.
Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties – those that become trivial after pullback by a finite étale map – are important; this acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures by Pierre Deligne. From 1959 onward Serre's interests turned towards group theory, number theory, in particular Galois representations and modular forms. Amongst his most original contributions were: his "Conjecture II" on Galois cohomology. In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free; this question led to a great deal of activity in commutative algebra, was answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976.
This result is now known as the Quillen–Suslin theorem. Serre, at twenty-seven in 1954, is the youngest to be awarded the Fields Medal, he went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000, was the first recipient of the Abel Prize in 2003. He has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre, he has received many honorary degrees. In 2012 he became a fellow of the American Mathematical Society. Serre has been awarded the highest honors in France as Grand Cross of the Legion of Honour and Grand Cross of the Legion of Merit. List of things named after Jean-Pierre Serre Nicolas Bourbaki Groupes Algébriques et Corps de Classes, translated into English as Algebraic Groups and Class Fields, Springer-Verlag Corps Locaux, Hermann, as Local Fields, Springer-Verlag Cohomologie Galoisienne Collège de France course 1962–63, as Galois Cohomology, Springer-Verlag Algèbre Locale, Multiplicités Collège de France course 1957–58, as Local Algebra, Springer-Verlag "Lie algebras and Lie groups" Harvard Lectures, Springer-Verlag.
Algèbres de Lie Semi-simples Complexes, as Complex Semisimple Lie Algebras, Springer-Verlag Abelian ℓ-Adic Representations and Elliptic Curves, CRC Press, reissue. Addison-Wesley. 1989. Cours d'arithmétique, PUF, as A Course in Arithmetic, Springer-Verlag Représentations linéaires des groupes finis, Hermann, as Linear Represent
Ernst Witt was a German mathematician, one of the leading algebraists of his time. Witt was born on the island of Alsen a part of the German Empire. Shortly after his birth, his parents moved the family to China, he did not return to Europe until he was nine. After his schooling, Witt went to the University of Göttingen, he was an active party member. Witt completed his Ph. D. at the University of Göttingen in 1934 with Emmy Noether and became a lecturer. He was a member of a team led by Helmut Hasse. During World War II he joined a group of five mathematicians, recruited by Wilhelm Fenner, which included Georg Aumann, Alexander Aigner, Oswald Teichmüller, Johann Friedrich Schultze and their leader professor Wolfgang Franz, to form the backbone of the new mathematical research department in the late 1930s, which would be called: Section IVc of Cipher Department of the High Command of the Wehrmacht. From 1937 until 1979, he taught at the University of Hamburg, he died in Hamburg in 1991, shortly after his 80th birthday.
Witt's work has been influential. His invention of the Witt vectors generalizes the structure of the p-adic numbers, it has become fundamental to p-adic Hodge theory. Witt was the founder of the theory of quadratic forms over an arbitrary field, he proved several of the key results, including the Witt cancellation theorem. He defined the Witt ring of all quadratic forms over a field, now a central object in the theory; the Poincaré–Birkhoff–Witt theorem is basic to the study of Lie algebras. In algebraic geometry, the Hasse–Witt matrix of an algebraic curve over a finite field determines the cyclic étale coverings of degree p of a curve in characteristic p. In the 1970s, Witt claimed that in 1940 he had discovered what would be named the "Leech lattice" many years before John Leech discovered it in 1965, but Witt did not publish his discovery and the details of what he did are unclear. List of things named after Ernst Witt Kersten, Ina, "Ernst Witt 1911-1991", Jahresbericht der Deutschen Mathematiker-Vereinigung, 95: 166–180 Witt, Kersten, Ina, ed. Collected papers.
Gesammelte Abhandlungen, New York: Springer-Verlag, ISBN 978-3-540-57061-5, MR 1643949 O'Connor, John J.. "Ernst Witt", MacTutor History of Mathematics archive, University of St Andrews. Ernst Witt at the Mathematics Genealogy Project
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the four mathematicians to have won the Fields Medal, the Wolf Prize, the Abel Prize. Milnor was born on February 1931 in Orange, New Jersey, his father was J. Willard Milnor and his mother was Emily Cox Milnor; as an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and proved the Fary–Milnor theorem. He continued on to graduate school at Princeton under the direction of Ralph Fox and submitted his dissertation, entitled "Isotopy of Links", which concerned link groups and their associated link structure, in 1954. Upon completing his doctorate he went on to work at Princeton, he was a professor at the Institute for Advanced Study from 1970 to 1990. His students have included Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. Siebenmann, Michael Spivak, his wife, Dusa McDuff, is a professor of mathematics at Barnard College.
One of his published works is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure. With Michel Kervaire, he showed that the 7-sphere has 15 differentiable structures. An n-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently Milnor worked on the topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fiber has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area which continues to mature to this day. In 1961 Milnor disproved the Hauptvermutung by illustrating two simplicial complexes which are homeomorphic but combinatorially distinct.
In 1984 Milnor introduced a definition of attractor. The objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors. Milnor's current interest is dynamics holomorphic dynamics, his work in dynamics is summarized by Peter Makienko in his review of Topological Methods in Modern Mathematics: It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the beginning, looking at the simplest nontrivial families of maps; the first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. The case of a unimodal map, that is, one with a single critical point, turns out to be rich; this work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems.
Milnor's work has opened several new directions in this field, has given us many basic concepts, challenging problems and nice theorems. He was an editor of the Annals of Mathematics for a number of years after 1962, he has written a number of books. In 1962 Milnor was awarded the Fields Medal for his work in differential topology, he went on to win the National Medal of Science, the Lester R. Ford Award in 1970 and again in 1984, the Leroy P Steele Prize for "Seminal Contribution to Research", the Wolf Prize in Mathematics, the Leroy P Steele Prize for Mathematical Exposition, the Leroy P Steele Prize for Lifetime Achievement "... for a paper of fundamental and lasting importance, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64, 399–405". In 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday. Milnor was awarded the 2011 Abel Prize, for his "pioneering discoveries in topology and algebra." Reacting to the award, Milnor told the New Scientist "It feels good," adding that "ne is always surprised by a call at 6 o'clock in the morning."
In 2013 he became a fellow of the American Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology and dynamical systems". Milnor, John W.. Morse theory. Annals of Mathematics Studies, No. 51. Notes by M. Spivak and R. Wells. Princeton, NJ: Princeton University Press. ISBN 0-691-08008-9. ——. Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton, NJ: Princeton University Press. ISBN 0-691-07996-X. OCLC 58324. ——. Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton, NJ: Princeton University Press. ISBN 0-691-08065-8. ——. Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08101-4. Husemoller, Dale. Symmetric bilinear forms. New York, NY: Springer-Verlag. ISBN 978-0-387-06009-5. Milnor, John W.. Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton, NJ: Princeton University Press. ISBN 0-691-08122-0. Milnor, John W..
Topology from the differentiable viewpoint. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-04833-9. —— (
Tsit Yuen Lam
Tsit Yuen Lam is a Hong Kong-American mathematician specializing in algebra ring theory and quadratic forms. Lam earned his bachelor's degree at the University of Hong Kong in 1963 and his Ph. D. at Columbia University in 1967 under Hyman Bass, with a thesis titled On Grothendieck Groups. Subsequently he was an instructor at the University of Chicago and since 1968 he has been at the University of California, where he became assistant professor in 1969, associate professor in 1972, full professor in 1976, he served as assistant department head several times. From 1995 to 1997 he was Deputy Director of the Mathematical Sciences Research Institute in Berkeley, California. Among his doctoral students is Richard Elman. From 1972 to 1974 he was a Sloan Fellow. In 1982 he was awarded the Leroy P. Steele Prize for his textbooks. In 2012 he became a fellow of the American Mathematical Society. Serre’s Conjecture. Lecture Notes in Mathematics, Springer, 1978 Serre’s Problem on Projective Modules. Springer 2006.
Benjamin 1973, 1980. Graduate Texts in Mathematics, Springer 1991, 2nd edition 2001, ISBN 0-387-95325-6 Lectures on Modules and Rings. Springer, Graduate Texts in Mathematics 1999, ISBN 978-0-387-98428-5 Orderings and Quadratic Forms. AMS 1983 Exercises in Classical Ring Theory. Springer 1985 Representations of Finite Groups: A Hundred Years. Part I, Part II. Notices of the AMS 1998. Lam's homepage
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local class field theory and diophantine geometry, to local zeta functions. Hasse was born in Kassel, Province of Hesse-Nassau, the son of Judge Paul Reinhard Hasse aka Haße and his wife Margarethe Louise Adolphine Quentin. After serving in the Imperial German Navy in World War I, he studied at the University of Göttingen, at the University of Marburg under Kurt Hensel, writing a dissertation in 1921 containing the Hasse–Minkowski theorem, as it is now called, on quadratic forms over number fields, he held positions at Kiel and Marburg. He was Hermann Weyl's replacement at Göttingen in 1934. Hasse was an Invited Speaker of the ICM in 1932 in Zurich and a Plenary Speaker of the ICM in 1936 in Oslo. In 1933 Hasse had signed the Loyalty Oath of German Professors to Adolf Hitler and the National Socialist State. Politically, he applied for membership in the Nazi Party in 1937, but this was denied to him due to his Jewish ancestry.
After the war, he returned to Göttingen in 1945, but was excluded by the British authorities. After brief appointments in Berlin, from 1948 on he settled permanently as professor in Hamburg, he collaborated with many mathematicians, in particular with Emmy Noether and Richard Brauer on simple algebras, with Harold Davenport on Gauss sums, with Cahit Arf on the Hasse–Arf theorem. Mathematische Abhandlungen, H. W. Leopoldt, Peter Roquette, 3 vols. de Gruyter 1975 Number theory, Springer, 1980, 2002 Vorlesungen über Zahlentheorie, Springer, 1950 Über die Klassenzahl abelscher Zahlkörper, Akademie Verlag, Berlin, 1952. Höhere Algebra vols. 1, 2, Sammlung Göschen, 1967, 1969 Vorlesungen über Klassenkörpertheorie, physica Verlag, Würzburg 1967 Hasse, H. "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie.", Jahresbericht der Deutschen Mathematiker-Vereinigung, 35: 1–55 Hasse, H. "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper.
Teil Ia: Beweise zu I.", Jahresbericht der Deutschen Mathematiker-Vereinigung, 36: 233–311 Hasse, H. "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz", Jahresbericht der Deutschen Mathematiker-Vereinigung, Ergänzungsband 6 Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, 1965 New edn. of Algebraische Theorie der Körper by Ernst Steinitz, together with Reinhold Baer, with a new appendix on Galois theory. Walter de Gruyter 1930. Hasse Mathematik als Wissenschaft, Kunst und Macht, DMV Mitteilungen 1997, Nr.4 Hasse „Geschichte der Klassenkörpertheorie“, Jahresbericht DMV 1966 Hasse „Die moderne algebraische Methode“, Jahresbericht DMV 1930 Brauer, Noether „Beweis eines Hauptsatzes in der Theorie der Algebren“, Journal reine angew. Math. 1932 Hasse „Theorie der abstrakten elliptischen Funktionenkörper 3- Riemann Vermutung“, Journal reine angew. Math. 1936 Hasse „Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen“, Journal reine angew.
Math. 1923 Hasse diagram Hasse invariant of an elliptic curve Hasse invariant of a quadratic form Artin–Hasse exponential Hasse–Weil L-function Hasse norm theorem Hasse's algorithm Hasse's theorem on elliptic curves O'Connor, John J.. "Helmut Hasse", MacTutor History of Mathematics archive, University of St Andrews. Another biography