HOME





Principal Root Of Unity
In mathematics, a principal ''n''-th root of unity (where ''n'' is a positive integer) of a ring is an element \alpha satisfying the equations : \begin & \alpha^n = 1 \\ & \sum_^ \alpha^ = 0 \text 1 \leq k < n \end In an , every primitive ''n''-th is also a principal n-th root of unity. In any ring, if ''n'' is a , then any ''n''/2-th root of −1 is a principal ''n''-th root of unity. A non-example is 3 in the
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

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]  


picture info

Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ...
[...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]  


picture info

Primitive Root Of Unity
In mathematics, a root of unity is any complex number that yields 1 when 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. If the 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, 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 number satisfying the equation z^n = 1. Unless otherwise specified, the roots ...
[...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]  


picture info

Power Of 2
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^n(1). In the Hardy hierarchy, is exactly equal to H_(1). Powers of two with non-negative exponents are integers: , , and is two multiplied by itself times. The first ten powers of 2 for non-negative values of are: : 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... By comparison, powers of two with negative exponents are fractions: for positive integer , is one half multiplied by itself times. Thus the first few negative powers of 2 are , , , , etc. Sometimes these are called ''inverse powers of two'' because each is the multiplicative inverse of a positive power of two. Base of the binary numeral system Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Ring Of Integers Modulo N
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This can be ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Cube Root Of Unity
In mathematics, a root of unity is any complex number that yields 1 when 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. If the 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, 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 number satisfying the equation z^n = 1. Unless otherwise specified, the roots o ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  




Discrete Fourier Transform (general)
In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring. Definition Let be any ring, let n\geq 1 be an integer, and let \alpha \in R be a principal ''n''th root of unity, defined by:Martin Fürer,Faster Integer Multiplication, STOC 2007 Proceedings, pp. 57–66. Section 2: The Discrete Fourier Transform. The discrete Fourier transform maps an ''n''-tuple (v_0,\ldots,v_) of elements of to another ''n''-tuple (f_0,\ldots,f_) of elements of according to the following formula: By convention, the tuple (v_0,\ldots,v_) is said to be in the ''time domain'' and the index is called ''time''. The tuple (f_0,\ldots,f_) is said to be in the ''frequency domain'' and the index is called ''frequency''. The tuple (f_0,\ldots,f_) is also called the ''spectrum'' of (v_0,\ldots,v_). This terminology derives from the applications of Fourier transforms ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Algebraic Numbers
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is an algebraic number, because it is a root of the polynomial X^2 - X - 1, i.e., a solution of the equation x^2 - x - 1 = 0, and the complex number 1 + i is algebraic as a root of X^4 + 4. Algebraic numbers include all integers, rational numbers, and ''n''-th roots of integers. Algebraic complex numbers are closed under addition, subtraction, multiplication and division, and hence form a field, denoted \overline. The set of algebraic real numbers \overline \cap \R is also a field. Numbers which are not algebraic are called transcendental and include and . There are countably many algebraic numbers, hence almost all real (or complex) numbers (in the sense of Lebesgue measure) are transcendental. Examples * All rational numbers are algebraic. Any rational number, express ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Cyclotomic Fields
In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime n)—and more precisely, because of the failure of unique factorization in their rings of integers—that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences. Definition For n \geq 1, let :\zeta_n=e^\in\C. This is a primitive nth root of unity. Then the nth cyclotomic field is the field extension \mathbb(\zeta_n) of \mathbb generated by \zeta_n. Properties * The nth cyclotomic polynomial :: \Phi_n(x) = \prod_\stackrel\!\!\! \left(x-e^\right) = \prod_\stackrel\!\!\! (x-^k) :is irreducible, so it is the minimal polynomial of \zeta_n over \Q. ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]