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Eisenstein's Criterion
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients. This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases for irreducibility to be proved with very little effort. It may apply either directly or after transformation of the original polynomial. This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it. Criterion Suppose we have the following polynomial with integer coefficients: Q(x) = a_n x^n + a_ x^ + \cdots + a_1 x + a_0. If there exists a prime number such that the following three conditions all apply: * divid ... [...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] |
Characteristic (algebra)
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said to have characteristic zero. That is, is the smallest positive number such that: : \underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: : \underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). This definition applies in the more general class of rngs (see '); for (unital) rings the two definitions are equivalent due to their distributive law. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive P-adic Valuation
In number theory, the valuation or -adic order of an integer is the exponent of the highest power of the prime number that divides . It is denoted \nu_p(n). Equivalently, \nu_p(n) is the exponent to which p appears in the prime factorization of n. The -adic valuation is a valuation and gives rise to an analogue of the usual absolute value. Whereas the completion of the rational numbers with respect to the usual absolute value results in the real numbers \mathbb, the completion of the rational numbers with respect to the p-adic absolute value results in the numbers \mathbb_p. Definition and properties Let be a prime number. Integers The -adic valuation of an integer n is defined to be : \nu_p(n)= \begin \mathrm\ & \text n \neq 0\\ \infty & \text n=0, \end where \mathbb_0 denotes the set of natural numbers (including zero) and m \mid n denotes divisibility of n by m. In particular, \nu_p is a function \nu_p \colon \mathbb \to \mathbb_0 \cup\ . For example, \nu_2(-12) = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lower Convex Envelope
In mathematics, the lower convex envelope \breve f of a function f defined on an interval ,b/math> is defined at each point of the interval as the supremum of all convex functions that lie under that function, i.e. : \breve f (x) = \sup\. See also * Convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ... * Lower envelope Convex analysis {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Polygon
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the ultrametric field of interest was ''essentially'' the field of formal Laurent series in the indeterminate ''X'', i.e. the field of fractions of the formal power series ring K X, over K, where K was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading terms aX^r of the power series expansion solutions to equations P(F(X)) = 0 where P is a polynomial with coefficients in K /math>, the polynomial ring; that is, implicitly defined algebraic functions. The exponents r here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in K Y with Y = X^ for a denominator d corresponding to the branch. The Newton polygon gives an e ... [...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 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'' which is not a unit can be written as a finite product of irreducible elements ''p''''i'' of ''R'': : ''x'' = ''p''1 ''p''2 � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss's Lemma (polynomial)
In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a theoremThis theorem is called a lemma for historical reasons. about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic). Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials. Gauss's lemma asserts that the product of two primitive polynomials is primitive. (A polynomial with integer coefficients is ''primitive'' if it has 1 as a greatest common divisor of its coefficients.The indefinite article is used here since, when the coefficients belong to a unique factorization domain, "greatest" refers to the preorder of divisibility, rather than to the natural order of the integers, and, generally, there are several greatest common divisors.) A corollary of Gauss's lemma, sometimes also called ''Gauss's lemma'', is that a primi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker as having said, "'" ("God made the integers, all else is the work of man").The English translation is from Gray. In a footnote, Gray attributes the German quote to "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886". Weber, Heinrich L. 1891–1892Kronecker 2:5-23. (The quote is on p. 19.) Kronecker was a student and life-long friend of Ernst Kummer. Biography Leopold Kronecker was born ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disquisitiones Arithmeticae
(Latin for ''Arithmetical Investigations'') is a textbook on number theory written in Latin by Carl Friedrich Gauss in 1798, when Gauss was 21, and published in 1801, when he was 24. It had a revolutionary impact on number theory by making the field truly rigorous and systematic and paved the path for modern number theory. In this book, Gauss brought together and reconciled results in number theory obtained by such eminent mathematicians as Fermat, Euler, Lagrange, and Legendre, while adding profound and original results of his own. Scope The ''Disquisitiones'' covers both elementary number theory and parts of the area of mathematics now called algebraic number theory. Gauss did not explicitly recognize the concept of a group, which is central to modern algebra, so he did not use this term. His own title for his subject was Higher Arithmetic. In his Preface to the ''Disquisitiones'', Gauss describes the scope of the book as follows: Gauss also writes, "When confronting man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemniscate Of Bernoulli
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from , which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4. This lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed ''focal points'' is a constant. A Cassini oval, by contrast, is the locus of points for which the ''product'' of these distances is constant. In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli. This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola (bisect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf[i] or \Z[i]. Gaussian integers share many properties with integers: they form a Euclidean domain, and thus have a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total order that respects arithmetic. Gaussian integers are algebraic integers and form the simplest ring of quadratic integers. Gaussian integers are named after the German mathematician Carl Friedrich Gauss. Basic definitions The Gaussian integers are the set :\mathbf[i]=\, \qquad \text i^2 = -1. In other words, a Gaussian integer is a complex number such that its real part, real and imaginary parts are both integers. Since the Gaussian integers are closed under addition a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
