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Abel's Irreducibility Theorem
In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if ''ƒ''(''x'') is a polynomial over a field ''F'' that shares a root with a polynomial ''g''(''x'') that is irreducible over ''F'', then every root of ''g''(''x'') is a root of ''ƒ''(''x''). Equivalently, if ''ƒ''(''x'') shares at least one root with ''g''(''x'') then ''ƒ'' is divisible evenly by ''g''(''x''), meaning that ''ƒ''(''x'') can be factored as ''g''(''x'')''h''(''x'') with ''h''(''x'') also having coefficients in ''F''..This theorem, for minimal polynomials rather than irreducible polynomials more generally, is Lemma 4.1.3 of . Irreducible polynomials, divided by their leading coefficient, are minimal for their roots (Cox Proposition 4.1.5), and all minimal polynomials are irreducible, so Cox's formulation is equivalent to Abel's. . Corollaries of the theorem include: * If ''ƒ''(''x'') is irreducible, ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Theory (mathematics) , a sociological theory concerning the relationship between social actors and local social orders
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Field theory may refer to: Science * Field (mathematics), the theory of the algebraic concept of field * Field theory (physics), a physical theory which employs fields in the physical sense, consisting of three types: ** Classical field theory, the theory and dynamics of classical fields ** Quantum field theory, the theory of quantum mechanical fields ** Statistical field theory, the theory of critical phase transitions **Grand unified theory Social science * Field theory (psychology), a psychological theory which examines patterns of interaction between the individual and his or her environment * Field theory (sociology) In sociology, field theory examines how individuals construct social fields, and how they are affected by such fields. Social fields are environments in which competition between individuals and between groups takes place, such as markets, academi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solving the general quintic equation in radicals. This question was one of the outstanding open problems of his day, and had been unresolved for over 250 years. He was also an innovator in the field of elliptic functions, discoverer of Abelian functions. He made his discoveries while living in poverty and died at the age of 26 from tuberculosis. Most of his work was done in six or seven years of his working life. Regarding Abel, the French mathematician Charles Hermite said: "Abel has left mathematicians enough to keep them busy for five hundred years." Another French mathematician, Adrien-Marie Legendre, said: "What a head the young Norwegian has!" The Abel Prize in mathematics, originally proposed in 1899 to complement the Nobel Prizes (but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crelle's Journal
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Rainer Weissauer ( Ruprecht-Karls-Universität Heidelberg) Past editors * 1826–1856 August Leopold Crelle * 1856–1880 Carl Wilhelm Borchardt * 1881–1888 Leopold Kronecker, Karl Weierstrass * 1889–1892 Leopold Kronecker * 1892–1902 Lazarus Fuchs * 1903–192 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' joins tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of A Polynomial
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is the solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6 has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real numbers, then ... [...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 field 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 po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves variables, they may also be called parameters. For example, the polynomial 2x^2-x+3 has coefficients 2, −1, and 3, and the powers of the variable x in the polynomial ax^2+bx+c have coefficient parameters a, b, and c. The constant coefficient is the coefficient not attached to variables in an expression. For example, the constant coefficients of the expressions above are the number 3 and the parameter ''c'', respectively. The coefficient attached to the highest degree of the variable in a polynomial is referred to as the leading coefficient. For example, in the expressions above, the leading coefficients are 2 and ''a'', respectively. Terminology and definition In mathematics, a coefficient is a multiplicative factor in some term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Polynomial (field Theory)
In field theory, a branch of mathematics, the minimal polynomial of an element of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the field, such that is a root of the polynomial. If the minimal polynomial of exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1, and the type for the remaining coefficients could be integers, rational numbers, real numbers, or others. More formally, a minimal polynomial is defined relative to a field extension and an element of the extension field . The minimal polynomial of an element, if it exists, is a member of , the ring of polynomials in the variable with coefficients in . Given an element of , let be the set of all polynomials in such that . The element is called a root or zero of each polynomial in . The set is so named because it is an ideal of . The zero polynomial, all of whose coefficients are 0, is in every since for all and . Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corollaries
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another proposition; it might also be used more casually to refer to something which naturally or incidentally accompanies something else (e.g., violence as a corollary of revolutionary social changes). Overview In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term ''corollary'', rather than ''proposition'' or ''theorem'', is intrinsically subjective. More formally, proposition ''B'' is a corollary of proposition ''A'', if ''B'' can be readily deduced from ''A'' or is self-evident from its proof. In many cases, a corollary corresponds to a special case of a larger theorem, which makes the theorem easier to use and apply, even though its importance is generally considered to be secondary t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
