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Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension n − 1, which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve
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Geometry
Geometry
Geometry
(from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry
Geometry
arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes
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Number Field
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4 and so forth.[1] A notational symbol that represents a number is called a numeral.[2] In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, number may refer to a symbol, a word, or a mathematical abstraction. In mathematics, the notion of number has been extended over the centuries to include 0,[3] negative numbers,[4] rational numbers such as 1/2 and −2/3, real numbers[5] such as √2 and π, and complex numbers,[6] which extend the real numbers by adding a square root of −1.[4] Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic
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Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring similarly to the way that, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may be distinct from the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring
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Square-free Polynomial
In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, a unique factorization domain) that does not have as a factor any square of a non-unit factor. In the important case of univariate polynomials over a field k, this means that f ∈ k [ X ] displaystyle fin k[X] is square-free if and only if b 2 ∤ f displaystyle b^ 2 nmid f for every polynomial b ∈ k [ X ] displaystyle bin k[X] of positive degree.[1] In applications in physics and engineering, a square-free polynomial is commonly called a polynomial with no repeated roots
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Height (ring Theory)
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension has been introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal I in a polynomial ring R is the Krull dimension of R/I. A field k has Krull dimension 0; more generally, k[x1, ..., xn] has Krull dimension n. A principal ideal domain that is not a field has Krull dimension 1
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Principal Ideal
In the mathematical field of ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x of P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept.Contents1 Definitions 2 Examples of non-principal ideal 3 Examples of principal ideal 4 Related definitions 5 Properties 6 See also 7 ReferencesDefinitions[edit]a left principal ideal of R is a subset of R of the form Ra = ra : r in R ; a right principal ideal is a subset of the form aR = ar : r in R ; a two-sided principal ideal is a subset of all finite sums of elements of the form ras, namely, RaR = r1as1 + ..
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Real Number
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as π (3.14159265...). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one
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Algebraically Closed Field
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F[x], the ring of polynomials in the variable x with coefficients in F.Contents1 Examples 2 Equivalent properties2.1 The only irreducible polynomials are those of degree one 2.2 Every polynomial is a product of first degree polynomials 2.3 Polynomials of prime degree have roots 2.4 The field has no proper algebraic extension 2.5 The field has no proper finite extension 2.6 Every endomorphism of Fn has some eigenvector 2.7 Decomposition of rational expressions 2.8 Relatively prime polynomials and roots3 Other properties 4 Notes 5 ReferencesExamples[edit] As an example, the field of real numbers is not algebraically closed, because the polynomial equation x2 + 1 = 0  has no solution in real numbers, even though all its coefficients (1 and 0) are real
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Rational Number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold Q displaystyle mathbb Q , Unicode ℚ);[2] it was thus denoted in 1895 by Giuseppe Peano
Giuseppe Peano
after quoziente, Italian for "quotient". The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g
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Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules
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Gaussian Rational
In mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers. The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(i), obtained by adjoining the imaginary number i to the field of rationals. Properties of the field[edit] The field of Gaussian rationals provides an example of an algebraic number field, which is both a quadratic field and a cyclotomic field (since i is a 4th root of unity). Like all quadratic fields it is a Galois extension of Q with Galois group cyclic of order two, in this case generated by complex conjugation, and is thus an abelian extension of Q, with conductor 4.[1] As with cyclotomic fields more generally, the field of Gaussian rationals is neither ordered nor complete (as a metric space). The Gaussian integers Z[i] form the ring of integers of Q(i)
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Algebraically Closed Extension
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F[x], the ring of polynomials in the variable x with coefficients in F.Contents1 Examples 2 Equivalent properties2.1 The only irreducible polynomials are those of degree one 2.2 Every polynomial is a product of first degree polynomials 2.3 Polynomials of prime degree have roots 2.4 The field has no proper algebraic extension 2.5 The field has no proper finite extension 2.6 Every endomorphism of Fn has some eigenvector 2.7 Decomposition of rational expressions 2.8 Relatively prime polynomials and roots3 Other properties 4 Notes 5 ReferencesExamples[edit] As an example, the field of real numbers is not algebraically closed, because the polynomial equation x2 + 1 = 0  has no solution in real numbers, even though all its coefficients (1 and 0) are real
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Homogeneous Polynomial
In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree.[1] For example, x 5 + 2 x 3 y 2 + 9 x y 4 displaystyle x^ 5 +2x^ 3 y^ 2 +9xy^ 4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x 3 + 3 x 2 y + z 7 displaystyle x^ 3 +3x^ 2 y+z^ 7 is not homogeneous, because the sum of exponents does not match from term to term. A polynomial is homogeneous if and only if it defines a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial.[2] A binary form is a form in two variables
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Monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:(1): A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. The constant 1 is a monomial, being equal to the empty product and x0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power xn of x, with n a positive integer
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Projective Coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius
August Ferdinand Möbius
in his 1827 work Der barycentrische Calcül,[1][2] are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates
Homogeneous coordinates
have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. If the homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point
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