|
Dedekind Domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below. A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields. An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the '' variety of groups''. History Before the nineteenth century, alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often denoted by O_K or \mathcal O_K. Since any integer belongs to K and is an integral element of K, the ring \mathbb is always a subring of O_K. The ring of integers \mathbb is the simplest possible ring of integers. Namely, \mathbb=O_ where \mathbb is the field of rational numbers. And indeed, in algebraic number theory the elements of \mathbb are often called the "rational integers" because of this. The next simplest example is the ring of Gaussian integers \mathbb /math>, consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field \mathbb(i) of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been known since antiquity to have infinitely many solutions.Singh, pp. 18–20. The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of ''Arithmetica''. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity and finite strain theory elaborate the mathematical abstractions). Biography Lamé was born in Tours, in today's ''département'' of Indre-et-Loire. He became well known for his general theory of curvilinear coordinates and his notation and study of classes of ellipse-like curves, now known as Lamé curves or superellipses, and defined by the equation: : \left, \,\,\^n + \left, \,\,\^n =1 where ''n'' is any positive real number. He is also known for his running time analysis of the Euclidean algorithm, marking the beginning of computational complexity theory. Using Fibonacci numbers, he proved that when finding the greatest common divisor of integers ''a'' and ''b'', the algorithm runs in no more than 5''k'' steps, where ''k'' is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \,\sim\, on S, the of an element a in S, denoted by is the set \ of elements which are equivalent to a. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S by \,\sim\,, and is denoted by S / \sim. When the set S has some structure (such as a group operation or a topology) and the equivalence relation \,\sim\, is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Stark
Harold Mead Stark (born August 6, 1939 in Los Angeles, California) is an American mathematician, specializing in number theory. He is best known for his solution of the Gauss class number 1 problem, in effect correcting and completing the earlier work of Kurt Heegner, and for Stark's conjecture. More recently, he collaborated with Audrey Terras to study zeta functions in graph theory. He is currently on the faculty of the University of California, San Diego. Stark received his bachelor's degree from California Institute of Technology in 1961 and his PhD from the University of California, Berkeley in 1964. He was on the faculty at the University of Michigan from 1964 to 1968, at the Massachusetts Institute of Technology from 1968 to 1980, and at the University of California, San Diego from 1980 to the present. Stark was elected to the American Academy of Arts and Sciences in 1983 and to the United States National Academy of Sciences in 2007. In 2012, he became a fellow of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Baker (mathematician)
Alan Baker (19 August 1939 – 4 February 2018) was an English mathematician, known for his work on effective methods in number theory, in particular those arising from transcendental number theory. Life Alan Baker was born in London on 19 August 1939. He attended Stratford Grammar School, East London, and his academic career started as a student of Harold Davenport, at University College London and later at Trinity College, Cambridge, where he received his PhD. He was a visiting scholar at the Institute for Advanced Study in 1970 when he was awarded the Fields Medal at the age of 31. In 1974 he was appointed Professor of Pure Mathematics at Cambridge University, a position he held until 2006 when he became an Emeritus. He was a fellow of Trinity College from 1964 until his death. His interests were in number theory, transcendence, logarithmic forms, effective methods, Diophantine geometry and Diophantine analysis. In 2012 he became a fellow of the American Mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kurt Heegner
Kurt Heegner (; 16 December 1893 – 2 February 1965) was a German private scholar from Berlin, who specialized in radio engineering and mathematics. He is famous for his mathematical discoveries in number theory and, in particular, the Stark–Heegner theorem. Life and career Heegner was born and died in Berlin. In 1952, he published the Stark–Heegner theorem which he claimed was the solution to a classic number theory problem proposed by the great mathematician Gauss, the class number 1 problem. Heegner's work was not accepted for years, mainly due to his quoting of a portion of Heinrich Martin Weber's work that was known to be incorrect (though he never used this result in the proof). Heegner's proof was accepted as essentially correct after a 1967 announcement by Bryan Birch, and definitively resolved by a paper by Harold Stark that had been delayed in publication until 1969 (Stark had independently arrived at a similar proof, but disagrees with the common notion that h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referred to as the ''Princeps mathematicorum'' () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and he is ranked among history's most influential mathematicians. Also available at Retrieved 23 February 2014. Comprehensive biographical article. Biography Early years Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Integer
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and (usual) integers. When algebraic integers are considered, the usual integers are often called ''rational integers''. Common examples of quadratic integers are the square roots of rational integers, such as , and the complex number , which generates the Gaussian integers. Another common example is the non-real cubic root of unity , which generates the Eisenstein integers. Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations, and other questions related to integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory. History Medieval Indian mathematicians had already discovered a multiplication of quadratic integers of the same , which allowed them to solve som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre De Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' '' Arithmetica''. He was also a lawyer at the '' Parlement'' of Toulouse, France. Biography Fermat was born in 1607 in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
