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Minimal Polynomial (field Theory)
In field theory, a branch of mathematics, the minimal polynomial of an element of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the smaller 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. 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 More specifically, ''J''''α'' is the kernel of the ring homomorphism from ''F'' 'x''to ''E'' which sends polynomials ''g'' to their value ''g''(''α'') at the element ''α''. Because it is the kernel of a ring homom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Theory (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. 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 field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''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 element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Element
In mathematics, if is an associative algebra over , then an element of is an algebraic element over , or just algebraic over , if there exists some non-zero polynomial g(x) \in K /math> with coefficients in such that . Elements of that are not algebraic over are transcendental over . A special case of an associative algebra over K is an extension field L of K. These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is , with being the field of complex numbers and being the field of rational numbers). Examples * The square root of 2 is algebraic over , since it is the root of the polynomial whose coefficients are rational. * Pi is transcendental over but algebraic over the field of real numbers : it is the root of , whose coefficients (1 and −) are both real, but not of any polynomial with only rational coefficients. (The definition of the term transcendental number uses , not .) Properties The following conditions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Polynomial (linear Algebra)
Minimal may refer to: * Minimal (music genre), art music that employs limited or minimal musical materials * "Minimal" (song), 2006 song by Pet Shop Boys * Minimal (supermarket) or miniMAL, a former supermarket chain in Germany and Poland * Minimal (''Dungeons & Dragons''), a creature of magically reduced size in the game ''Dungeons & Dragons'' * Minimal (chocolate), a bean to bar chocolate store in Japan, featured in '' Kantaro: The Sweet Tooth Salaryman'' * Minimal (clothing), an Indonesia clothing-retail company that worked with fashion model Ayu Gani * MINIMAL (restaurant), high end restaurant in Taichung Taichung (, Wade–Giles: '), officially Taichung City, is a special municipality (Taiwan), special municipality in central Taiwan. Taichung is Taiwan's second-largest city, with more than 2.85 million residents, making it the largest city in Ce ..., Taiwan See also * * Minimalism (other) * Maximal (other) * Minimisation (other) * Minimal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. These operations make the field into an abelian group under addition, and they make the nonzero elements of the field into another abelian group under multiplicat ... [...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, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Swinnerton-Dyer Polynomial
In algebra, the Swinnerton-Dyer polynomials are a family of polynomials, introduced by Peter Swinnerton-Dyer, that serve as examples where polynomial factorization algorithms have worst-case runtime. They have the property of being reducible modulo every prime, while being irreducible over the rational numbers. They are a standard counterexample in number theory. Given a finite set P of prime numbers, the Swinnerton-Dyer polynomial associated to P is the polynomial: f_P(x) = \prod \left(x + \sum_ (\pm) \sqrt\right) where the product extends over all 2^ choices of sign in the enclosed sum. The polynomial f_P(x) has degree 2^ and integer coefficients, which alternate in sign. If , P, >1, then f_P(x) is reducible modulo p for all primes p, into linear and quadratic factors, but irreducible over \mathbb Q. The Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Polynomial Of 2cos(2pi/n)
In number theory, the real parts of the roots of unity are related to one-another by means of the minimal polynomial of 2\cos(2\pi/n). The roots of the minimal polynomial are twice the real part of the roots of unity, where the real part of a root of unity is just \cos\left(2k\pi/n\right) with k coprime with n. Formal definition For an integer n \geq 1, the minimal polynomial \Psi_n(x) of 2\cos(2\pi/n) is the non-zero integer-coefficient monic polynomial of smallest degree for which \Psi_n\!\left(2\cos(2\pi/n)\right) = 0. For every , the polynomial \Psi_n(x) is monic, has integer coefficients, and is irreducible over the integers and the rational numbers. All its roots are real; they are the real numbers 2\cos\left(2k\pi/n\right) with k coprime with n and either 1 \le k 1 of n, including n itself: :\prod _\Psi_( x)=\chi_( x). This means that the \Psi_(x) are exactly the irreducible factors of \chi_(x), which allows to easily obtain \Psi_(x) for any odd d, knowing its degree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclotomic Polynomial
In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primitive roots of unity e^ , where ''k'' runs over the positive integers less than ''n'' and coprime to ''n'' (and ''i'' is the imaginary unit). In other words, the ''n''th cyclotomic polynomial is equal to : \Phi_n(x) = \prod_\stackrel \left(x-e^\right). It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive ''n''th-root of unity ( e^ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is :\prod_\Phi_d(x) = x^n - 1, showing that x is a root of x^n - 1 if and only if it is a ''d''th primitive root of unity for some ''d'' that divides ''n''. Examples If ''n'' is a prim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of Unity
In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre. Roots of unity can be defined in any field (mathematics), field. If the characteristic of a field, characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, converse (logic), conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a nu ... [...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. A complex number is called a quadratic integer if it is a root of some monic polynomial (a polynomial whose leading coefficient is 1) of degree two whose coefficients are integers, i.e. quadratic integers are algebraic integers of degree two. Thus quadratic integers are those complex numbers that are 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 \sqrt, and the complex number i=\sqrt, which generates the Gaussian integers. Another common example is the non- real cubic root of unity \frac, 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 quadrati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Ideal Domain
In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some authors such as Bourbaki refer to PIDs as principal rings. Principal ideal domains are mathematical objects that behave like the integers, with respect to divisibility: any element of a PID has a unique factorization into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If and are elements of a PID without common divisors, then every element of the PID can be written in the form , etc. Principal ideal domains are Noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Domain
In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element ''a'' has the cancellation property, that is, if , an equality implies . "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using " domain" for the general case including noncommutative rings. Some sources, notably Lang, use the term entire ring for integral domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
