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Latimer–MacDuffee Theorem
The Latimer–MacDuffee theorem is a theorem in abstract algebra, a branch of mathematics. It is named after Claiborne Latimer and Cyrus Colton MacDuffee, who published it in 1933. Significant contributions to its theory were made later by Olga Taussky-Todd.. Let f be a monic, irreducible polynomial of degree n. The Latimer–MacDuffee theorem gives a one-to-one correspondence between \mathbb- similarity classes of n\times n matrices with characteristic polynomial f and the ideal classes in the order :\mathbb (f(x)). where ideals are considered equivalent if they are equal up to an overall (nonzero) rational scalar multiple. (Note that this order need not be the full ring of integers, so nonzero ideals need not be invertible.) Since an order in a number field has only finitely many ideal classes (even if it is not the maximal order, and we mean here ideals classes for all nonzero ideals, not just the invertible ones), it follows that there are only finitely many conjugacy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formal system ... [...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, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claiborne Latimer
Claiborne Green Latimer (1893–1960) was an American mathematician, known for the Latimer–MacDuffee theorem. Career Latimer earned his PhD in 1924 from the University of Chicago under Leonard Dickson with thesis ''Arithmetic of Generalized Quaternion Algebras''. He was an assistant professor at Tulane University for 2 years, before becoming a mathematics professor at the University of Kentucky in 1927. After 20 years at the University of Kentucky, he resigned in 1947 and became a professor at Emory University Emory University is a private university, private research university in Atlanta, Georgia, United States. It was founded in 1836 as Emory College by the Methodist Episcopal Church and named in honor of Methodist bishop John Emory. Its main campu .... Latimer was an amateur photographer; some of his photographs are preserved in the archives of the University of Kentucky and Emory University. References 1893 births 1960 deaths 20th-century American mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyrus Colton MacDuffee
Cyrus Colton MacDuffee (June 29, 1895 – August 21, 1961) from Oneida, New York was a professor of mathematics at University of Wisconsin. He wrote a number of influential research papers in abstract algebra. MacDuffee served on the Council of the American Mathematical Society (A.M.S.), was editor of the '' Transactions of the A.M.S.'', and served as president of the Mathematical Association of America (M.A.A). MacDuffee obtained his B.S. degree in 1917 from Colgate University and a Ph.D. in 1922 from the University of Chicago; his thesis was on Nonassociative algebras under the direction of Leonard E. Dickson. In 1935, MacDuffee joined the University of Wisconsin, where he remained until his death in 1961. He served as chair of the department (1951–56). Later, Wisconsin endowed a university chair under his name. Prior to joining the University of Wisconsin, he served at Princeton and Ohio State. He guided 30 Ph.D. students, among them D. R. Fulkerson, H. J. Ryser, and Bonnie S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Olga Taussky-Todd
Olga Taussky-Todd (August 30, 1906 – October 7, 1995) was an Austrian and later Czech Americans, Czech-American mathematician. She published more than 300 research papers on algebraic number theory, integral matrices, and Matrix (mathematics), matrices in algebra and Analysis (math), analysis. Early life Olga Taussky was born into a Jewish family in what is now Olomouc, Czech Republic, on August 30, 1906. Her father, Julius David Taussky, was an industrial chemist and her mother, Ida Pollach, was a housewife. She was the second of three children. Her father preferred that, if his daughters had careers, they be in the arts, but they all went into the sciences. Ilona, three years older than Olga, became a consulting chemist in the glyceride industry, and Hertha, three years younger than Olga, became a pharmacist and later a clinical chemist at Cornell University Medical College in New York City. At the age of three, her family moved to Vienna and lived there until the middle of W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monic Polynomial
In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one that can be written as :x^n+c_x^+\cdots+c_2x^2+c_1x+c_0, with n \geq 0. Uses Monic polynomials are widely used in algebra and number theory, since they produce many simplifications and they avoid divisions and denominators. Here are some examples. Every polynomial is associated to a unique monic polynomial. In particular, the unique factorization property of polynomials can be stated as: ''Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials.'' Vieta's formulas are simpler in the case of monic polynomials: ''The th elementary symmetric function of the roots of a monic polynomial of degree equals (-1)^ic_, where c_ is the coefficient of the th po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as \left(x - \sqrt\right)\left(x + \sqrt\right) if it is considered as a polynomial with real coefficients. One says that the polynomial is irreducible over the integers but not over the reals. Polynomial irreducibility can be considered for polynomials with coefficients in an integral domain, and there are two common definitions. Most often, a pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Similarity
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that B = P^ A P . Similar matrices represent the same linear map under two possibly different bases, with being the change-of-basis matrix. A transformation is called a similarity transformation or conjugation of the matrix . In the general linear group, similarity is therefore the same as conjugacy, and similar matrices are also called conjugate; however, in a given subgroup of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that be chosen to lie in . Motivating example When defining a linear transformation, it can be the case that a change of basis can result in a simpler form of the same transformation. For example, the matrix representing a rotation in when the axis of rotation is not aligned with the coordinate axis can be complicated to compute. If the axis of rotation were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
