Quartic Field
In mathematics, the term quartic describes something that pertains to the "fourth order", such as the function x^4. It may refer to one of the following: * Quartic function, a polynomial function of degree 4 * Quartic equation, a polynomial equation of degree 4 * Quartic curve, an algebraic curve of degree 4 * Quartic reciprocity, a theorem from number theory * Quartic surface, a surface defined by an equation of degree 4 ; See also * * * Quart (other) * Quintic, relating to degree 5, as next higher above ''quartic'' * Cubic (other), relating to degree 3 or a cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

Quartic Function
In algebra, a quartic function is a function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A ''quartic equation'', or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form :ax^4+bx^3+cx^2+dx+e=0 , where . The derivative of a quartic function is a cubic function. Sometimes the term biquadratic is used instead of ''quartic'', but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form :f(x)=ax^4+cx^2+e. Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If ''a'' is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. Likewise, if ''a'' is nega ...
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Quartic Equation
In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomial equation that can be solved by radicals in the general case (i.e., one in which the coefficients can take any value). History Lodovico Ferrari is attributed with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it couldn't be published immediately. The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book '' Ars Magna'' (1545). The proof that this was the highest order general polynomial for which such solutions could be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would ...
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Quartic Curve
In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one of not equal to zero. This equation has 15 constants. However, it can be multiplied by any non-zero constant without changing the curve; thus by the choice of an appropriate constant of multiplication, any one of the coefficients can be set to 1, leaving only 14 constants. Therefore, the space of quartic curves can be identified with the real projective space It also follows, from Cramer's theorem on algebraic curves, that there is exactly one quartic curve that passes through a set of 14 distinct points in general position, since a quartic has 14 degrees of freedom. A quartic curve can have a maximum of: * Four connected components * Twenty-eight bi-tangents * Three ordinary double points. One may also consider quartic curves over o ...
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Quartic Reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence ''x''4 ≡ ''p'' (mod ''q'') is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence ''x''4 ≡ ''p'' (mod ''q'') to that of ''x''4 ≡ ''q'' (mod ''p''). History Euler made the first conjectures about biquadratic reciprocity. Gauss published two monographs on biquadratic reciprocity. In the first one (1828) he proved Euler's conjecture about the biquadratic character of 2. In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas. He saidGauss, BQ, § 67 that a third monograph would be forthcoming with the proof of the general theorem, but it never appeared. Jacobi presented proofs in his Königsberg lectures of 1836–37. The first published proofs were by Eise ...
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Quartic Surface
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine and projective. An ''affine'' quartic surface is the solution set of an equation of the form :f(x,y,z)=0\ where is a polynomial of degree 4, such as . This is a surface in affine space . On the other hand, a projective quartic surface is a surface in projective space of the same form, but now is a ''homogeneous'' polynomial of 4 variables of degree 4, so for example . If the base field is or the surface is said to be ''real'' or ''complex'' respectively. One must be careful to distinguish between algebraic Riemann surfaces, which are in fact quartic curves over , and quartic surfaces over . For instance, the Klein quartic is a ''real'' surface given as a quartic curve over . If on the other hand the base field is finite, then it is said to be an ''arithmetic quartic surfa ...
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Quart (other)
Quart is an imperial and US customary unit of volume. Quart may also refer to: * Quart, a coin worth of a Gibraltarian real * Quart Festival, a music festival in Norway * Quart, a proposed metric typographic unit Places * Quart, Aosta Valley, a ''comune'' in Italy * Quart, Girona, a municipality in Spain * Quart de les Valls, a municipality in Spain * Quart de Poblet, a municipality in Spain People * Alberto Arnal (1913–1966), also known as Quart, Valencian pilotari (handball player) * Emerico di Quart (died 1313), beatified Catholic bishop * Josie Alice Quart (1895–1980), member of the Canadian Senate See also * Quartz Quartz is a hard, crystalline mineral composed of silica (silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall chemical form ...

Quintic
In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a quintic function is defined by a polynomial of degree five. Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. The derivative of a quintic function is a quartic function. Setting and assuming produces a quintic equation of the form: :ax^5+bx^4+cx^3+dx^2+ex+f=0.\, Solving quintic equations in terms of radicals (''n''th roots) was a major problem in algebra from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the Abel–Ruffini theorem. Finding roots of a quintic equa ...
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Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one can ...
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Cubic (other)
Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system where the unit cell is in the shape of a cube * Cubic function, a polynomial function of degree three * Cubic equation, a polynomial equation (reducible to ''ax''3 + ''bx''2 + ''cx'' + ''d'' = 0) * Cubic form, a homogeneous polynomial of degree 3 * Cubic graph (mathematics - graph theory), a graph where all vertices have degree 3 * Cubic plane curve (mathematics), a plane algebraic curve ''C'' defined by a cubic equation * Cubic reciprocity (mathematics - number theory), a theorem analogous to quadratic reciprocity * Cubic surface, an algebraic surface in three-dimensional space * Cubic zirconia, in geology, a mineral that is widely synthesized for use as a diamond simulacra * CUBIC, a histology method Computing * Cubic IDE, a modular deve ...
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