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Euclidean Division Of Polynomials
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factorization, factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers. In the important case of univariate polynomials over a field (mathematics), field the polynomial GCD may be computed, like for the integer GCD, by the Euclidean algorithm using polynomial long division, long division. The polynomial GCD is defined only up to the multiplication by an invertible constant. The similarity between the integer GCD and the polynomial GCD allows extending to univariate polynomials all the properties that may be deduced from the Euclidean algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the root of a function, roots of the GCD of two polynomials are the common roots ...
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Polynomial
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a polynomial is an expression consisting of indeterminates (also called variables) and coefficient In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...s, that involves only the operations of addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ..., subtraction Subtraction is an arithmetic operation that repre ...
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Computer Algebra System
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software Mathematical software is software used to mathematical model, model, analyze or calculate numeric, symbolic or geometric data. It is a type of application software which is used for solving mathematical problems or mathematical study. There are v ... with the ability to manipulate mathematical expressions In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed formula, well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constant (mathematics ... in a way similar to the traditional manual computations of mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...s and scienti ...
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Real Number
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ..., a real number is a value of a continuous quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measu ... that can represent a distance along a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ... (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective ''real'' in this co ...
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Floating-point Numbers
In computing, floating-point arithmetic (FP) is arithmetic using formulaic representation of real numbers as an approximation to support a trade-off between range and precision. For this reason, floating-point computation is often used in systems with very small and very large real numbers that require fast processing times. In general, a floating-point number is represented approximately with a fixed number of Significant figures, significant digits (the significand) and scaled using an exponentiation, exponent in some fixed base; the base for the scaling is normally two, ten, or sixteen. A number that can be represented exactly is of the following form: \text \times \text^\text, where ''significand'' is an integer, ''base'' is an integer greater than or equal to two, and ''exponent'' is also an integer. For example: 1.2345 = \underbrace_\text \times \underbrace_\text\!\!\!\!\!\!^. The term ''floating point'' refers to the fact that a number's radix point (''decimal point'', or ...
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Finitely Generated Field Extension
In mathematics, particularly in algebra, a field extension is a pair of Field (mathematics), fields E\subseteq F, such that the operations of ''E'' are those of ''F'' Restriction (mathematics), restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ''F''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield of a field (mathematics), field ''L'' is a subset ''K'' of ''L'' that is a field with respect to the field operations inherited from ''L''. Equivalently, a subfield is a subset that contains 1, and is Closure (mathematics), closed under the operations of addition, subtraction, multiplication, and taking the multiplicativ ...
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Finite Field
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a finite field or Galois field (so-named in honor of Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...) is a field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ... that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addi ...
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Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set (mathematics), set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface (or blackboard bold \mathbb, Unicode or ); it was thus denoted in 1895 by Giuseppe Peano after ''wikt:quoziente, quoziente'', Italian for "quotient", and first appeared in Bourbaki's ''Algèbre''. The decimal expansion of a rational number either terminates after a finite number of numerical digit, digits (example: ), or eventually begins to repeating decimal, repeat the same finite sequence of digits over and over (example: ). Conversely, any repeating or terminating decimal represents a rational number. These statements are true in decimal, base 10, and in ev ...
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Integer
An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ... ''integer'' meaning "whole") is colloquially defined as a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ... that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not. The set of integers consists of zero (), the positive natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' larges ...
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Pseudo-remainder Sequences
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factorization, factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers. In the important case of univariate polynomials over a field (mathematics), field the polynomial GCD may be computed, like for the integer GCD, by the Euclidean algorithm using polynomial long division, long division. The polynomial GCD is defined only up to the multiplication by an invertible constant. The similarity between the integer GCD and the polynomial GCD allows extending to univariate polynomials all the properties that may be deduced from the Euclidean algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the root of a function, roots of the GCD of two polynomials are the common roots ...
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Factorization Of Polynomials
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ... and computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating expression (mathematics), ..., factorization of polynomials or polynomial factorization expresses a polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... with coefficients in a given field Field may refer to: Expanses of open ground * Field (agriculture), an area of ...
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Bézout's Identity
In elementary number theory, Bézout's identity (also called Bézout's lemma) is the following theorem: Here the greatest common divisor of 0 and 0 is taken to be 0. The integers ''x'' and ''y'' are called Bézout coefficients for (''a'', ''b''); they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is one of the two pairs such that , x, \le , b/d , and , y, \le , a/d , . An equality may occur only if one of ''a'' and ''b'' is a multiple of the other. As an example, the greatest common divisor of 15 and 69 is 3, and can be written . Many other theorems in elementary number theory, such as Euclid's lemma or Chinese remainder theorem, result from Bézout's identity. A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that results from Bézout's identity is thus true in all these domains. Structure of solutio ...
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Monic Polynomial
In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ..., a monic polynomial is a single-variable polynomial (that is, a univariate polynomial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...) in which the leading coefficient In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: : ...
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