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Square-free
{{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. Alternate characterizations Square-free elements may be also characterized using their prime decomposition. The unique factorization property means that a non-zero non-unit ''r'' can be represented as a product of prime elements :r=p_1p_2\cdots p_n Then ''r'' is square-free if and only if the primes ''pi'' are pairwise non-associated (i.e. that it doesn't have two of the same prime as factors, which would make it divisible by a square number). Examples Common examples of square-free elements include square-free integers and square-free polynomials. See also *Prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square-free Integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are Square-free factorization Every positive integer n can be factored in a unique way as n=\prod_^k q_i^i, where the q_i different from one are square-free integers that are pairwise coprime. This is called the ''square-free factorization'' of . To construct the square-free factorization, let n=\prod_^h p_j^ be the prime factorization of n, where the p_j are distinct prime numbers. Then the factors of the square-free factorization are defined as q_i=\prod_p_j. An integer is square-free if and only if q_i=1 for all i > 1. An integer greater than one is the kth power of another integer if and only if k is a divisor of all i such that q_i\neq 1. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square-free Polynomial
In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial. A univariate polynomial is square free if and only if it has no multiple root in an algebraically closed field containing its coefficients. This motivates that, in applications in physics and engineering, a square-free polynomial is commonly called a polynomial with no repeated roots. In the case of univariate polynomials, the product rule implies that, if divides , then divides the formal derivative of . The converse is also true and hence, f is square-free if and only if 1 is a greatest common divisor of the polynomial and its derivative. A square-free decomposition or square-free factorization of a polynomial is a factorization into powers of square-free polynomials : f = a_1 a_2^2 a_3^3 \cdots a_n^n =\prod_^n a_k^k\, where those of the that are non-constant are pairwise coprime squar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unique Factorization Domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units. Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field. Unique factorization domains appear in the following chain of class inclusions: Definition Formally, a unique factorization domain is defined to be an integral domain ''R'' in which every non-zero element ''x'' of ''R'' can be written as a product (an empty product if ''x'' is a unit) of irreducible elements ''p''i of ''R'' and a uni ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Domain
In mathematics, specifically abstract algebra, 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 entir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit (algebra)
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ). Less commonly, the term ''unit'' is sometimes used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng. Examples The multiplicative identity and its additive inverse are always units. More generally, any root of unity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Element
In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements, a concept which is the same in UFDs but not the same in general. Definition An element of a commutative ring is said to be prime if it is not the zero element or a unit and whenever divides for some and in , then divides or divides . With this definition, Euclid's lemma is the assertion that prime numbers are prime elements in the ring of integers. Equivalently, an element is prime if, and only if, the principal ideal generated by is a nonzero prime ideal. (Note that in an integral domain, the ideal is a prime ideal, but is an exception in the definition of 'prime element'.) Interest in prime elements comes from the fundamental theorem of arithmetic, which asserts that each nonzero in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Element
In mathematics, specifically abstract algebra, 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 ent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
