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Solution In Radicals
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of th roots ( square roots, cube roots, etc.). A well-known example is the quadratic formula :x=\frac, which expresses the solutions of the quadratic equation :ax^2 + bx + c =0. There exist algebraic solutions for cubic equations and quartic equations, which are more complicated than the quadratic formula. The Abel–Ruffini theorem,Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, and, more generally Galois theory, state that some quintic equations, such as :x^5-x+1=0, do not have any algebraic solution. The same is true for every higher degree. However, for any degree there are some polynomial equations that have algebraic solutions; for example, the equation x^ = 2 can be solved as x=\pm\sqrt 0. The eight other solutions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expression (mathematics)
In mathematics, an expression is a written arrangement of symbol (mathematics), symbols following the context-dependent, syntax (logic), syntactic conventions of mathematical notation. Symbols can denote numbers, variable (mathematics), variables, operation (mathematics), operations, and function (mathematics), functions. Other symbols include punctuation marks and bracket (mathematics), brackets, used for Symbols of grouping, grouping where there is not a well-defined order of operations. Expressions are commonly distinguished from ''mathematical formula, formulas'': expressions are a kind of mathematical object, whereas formulas are statements ''about'' mathematical objects. This is analogous to natural language, where a noun phrase refers to an object, and a whole Sentence (linguistics), sentence refers to a fact. For example, 8x-5 is an expression, while the Inequality (mathematics), inequality 8x-5 \geq 3 is a formula. To ''evaluate'' an expression means to find a numeric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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 could not 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 be futile. The notes left by Évariste Galois before ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Septic Equation
In algebra, a septic equation is an equation of the form :ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\, where . A septic function is a function of the form :f(x)=ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h\, where . In other words, it is a polynomial of degree seven. If , then ''f'' is a sextic function (), quintic function (), etc. The equation may be obtained from the function by setting . The ''coefficients'' may be either integers, rational numbers, real numbers, complex numbers or, more generally, members of any field. Because they have an odd degree, septic functions appear similar to quintic and cubic functions when graphed, except they may possess additional local maxima and local minima (up to three maxima and three minima). The derivative of a septic function is a sextic function. Solvable septics Some seventh degree equations can be solved by factorizing into radicals, but other septics cannot. Évariste Galois developed techniques for determining whether a given equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sextic Equation
In algebra, a sextic (or hexic) polynomial is a polynomial of degree six. A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form: :ax^6+bx^5+cx^4+dx^3+ex^2+fx+g=0,\, where and the ''coefficients'' may be integers, rational numbers, real numbers, complex numbers or, more generally, members of any field. A sextic function is a function defined by a sextic polynomial. Because they have an even degree, sextic functions appear similar to quartic functions when graphed, except they may possess an additional local maximum and local minimum each. The derivative of a sextic function is a quintic function. Since a sextic function is defined by a polynomial with even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If the leading coefficient is positive, then the function increases to positive infinity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Symbol
In mathematics, the radical symbol, radical sign, root symbol, or surd is a symbol for the square root or higher-order root of a number. The square root of a number is written as :\sqrt, while the th root of is written as :\sqrt It is also used for other meanings in more advanced mathematics, such as the radical of an ideal. In linguistics, the symbol is used to denote a root word. Principal square root Each positive real number has two square roots, one positive and the other negative. The radical symbol refers to the principal value of the square root function called the principal square root, which is the positive one. The two square roots of a negative number are both imaginary numbers, and the square root symbol refers to the principal square root, the one with a positive imaginary part. For the definition of the principal square root of other complex numbers, see . Origin The origin of the root symbol √ is largely speculative. Some sources imply that the symbol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Extension
In mathematics and more specifically in field theory, a radical extension of a field K is a field extension obtained by a tower of field extensions, each generated by adjoining an nth root of an element from the previous field. Definition A simple radical extension is a simple extension ''F''/''K'' generated by a single element \alpha satisfying \alpha^n = b for an element ''b'' of ''K''. In characteristic ''p'', we also take an extension by a root of an Artin–Schreier polynomial to be a simple radical extension. A radical series is a tower K = F_0 < F_1 < \cdots < F_k where each extension is a simple radical extension. In this case, the field extension is called a radical extension. Properties # If ''E'' is a radical extension of ''F'' and ''F'' is a radical extension of ''K,'' then ''E'' is a radical extension of ''K''. # If ''E'' and ''F'' are radical extensions of ''K'' in an |
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, radicals, thereby solving a problem that had been open for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra. Galois was a staunch French Republicans under the Restoration, republican and was heavily involved in the political turmoil that surrounded the French Revolution of 1830. As a result of his political activism, he was arrested repeatedly, serving one jail sentence of several months. For reasons that remain obscure, shortly after his release from prison, Galois fought in a duel and died of the wounds he suffered. Life Early life Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (née Demante). His father was a First French Republi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nested Radical
In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include \sqrt, which arises in discussing the pentagon, regular pentagon, and more complicated ones such as \sqrt[3]. Denesting Some nested radicals can be rewritten in a form that is not nested. For example, \sqrt = 1+\sqrt\,, \sqrt[3] = \frac \,. Another simple example, \sqrt[3] = \sqrt[6] Rewriting a nested radical in this way is called denesting. This is not always possible, and, even when possible, it is often difficult. Two nested square roots In the case of two nested square roots, the following theorem completely solves the problem of denesting. If and are rational numbers and is not the square of a rational number, there are two rational numbers and such that \sqrt = \sqrt\pm\sqrt if and only if a^2-c~ is the square of a rational number . If the nested radical is real, and are the two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of Unity
In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre. Roots of unity can be defined in any field (mathematics), field. If the characteristic of a field, characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, converse (logic), conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quintic Equation
In mathematics, 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 q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying root of a function, roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is by definition ''solvable by radicals'' if its roots may be expressed by a formula involving only integers, nth root, th roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
