65 relations: Abelian group, Abstract algebra, Algebraic equation, Algebraic independence, Algebraic solution, Alternating group, American Mathematical Monthly, Archive for History of Exact Sciences, Augustin-Louis Cauchy, Automorphism, Carl Friedrich Gauss, Characteristic (algebra), Coefficient, Complex number, Composition series, CRC Press, Cube root, Cubic function, Cyclic group, Disquisitiones Arithmeticae, Elementary symmetric polynomial, French Academy of Sciences, Fundamental theorem of algebra, Galois group, Galois theory, Grøndahl & Søn Forlag, Ian Stewart (mathematician), Icosahedral symmetry, Indeterminate (variable), Isomorphism, Joseph Liouville, Joseph-Louis Lagrange, Journal de Mathématiques Pures et Appliquées, Klein four-group, Laguerre's method, Michael Rosen (mathematician), MIT Press, Nathan Jacobson, Newton's method, Niels Henrik Abel, Normal subgroup, Nouvelles Annales de Mathématiques, Nth root, Paolo Ruffini, Permutation, Pierre Wantzel, Polynomial long division, Q.E.D., Quadratic formula, Quartic function, ..., Quotient group, Real number, Resolvent (Galois theory), Root of unity, Simple group, Solvable group, Splitting field, Springer Science+Business Media, Square root, Symmetric function, Thesis, Topological Galois theory, Topology, Vladimir Arnold, World Scientific. Expand index (15 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
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Algebraic equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form where P and Q are polynomials with coefficients in some field, often the field of the rational numbers.
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Algebraic independence
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. In particular, a one element set is algebraically independent over K if and only if α is transcendental over K. In general, all the elements of an algebraically independent set S over K are by necessity transcendental over K, and over all of the field extensions over K generated by the remaining elements of S.
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Algebraic solution
An algebraic solution or solution in radicals is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots (square roots, cube roots, and other integer roots).
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Alternating group
In mathematics, an alternating group is the group of even permutations of a finite set.
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American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.
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Archive for History of Exact Sciences
Archive for History of Exact Sciences is a peer-reviewed academic journal published quarterly by Springer Science+Business Media, covering the history of mathematics and of astronomy observations and techniques, epistemology of science, and philosophy of science from Antiquity until now.
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Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.
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Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself.
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Carl Friedrich Gauss
Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.
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Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
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Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Composition series
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces.
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CRC Press
The CRC Press, LLC is a publishing group based in the United States that specializes in producing technical books.
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Cube root
In mathematics, a cube root of a number x is a number y such that y3.
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Cubic function
In algebra, a cubic function is a function of the form in which is nonzero.
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Cyclic group
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
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Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24.
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Elementary symmetric polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials.
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French Academy of Sciences
The French Academy of Sciences (French: Académie des sciences) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research.
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Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
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Galois group
In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.
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Galois theory
In the field of algebra within mathematics, Galois theory, provides a connection between field theory and group theory.
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Grøndahl & Søn Forlag
Grøndahl & Søn Forlag was a Norwegian publishing house established in 1812.
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Ian Stewart (mathematician)
Ian Nicholas Stewart (born 24 September 1945) is a British mathematician and a popular-science and science-fiction writer.
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Icosahedral symmetry
A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation.
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Indeterminate (variable)
In mathematics, and particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else but itself and is used as a placeholder in objects such as polynomials and formal power series.
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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Joseph Liouville
Joseph Liouville FRS FRSE FAS (24 March 1809 – 8 September 1882) was a French mathematician.
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Joseph-Louis Lagrange
Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.
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Journal de Mathématiques Pures et Appliquées
The Journal de Mathématiques Pures et Appliquées is a French monthly scientific journal of mathematics, founded in 1836 by Joseph Liouville (editor: 1836–1874).
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Klein four-group
In mathematics, the Klein four-group (or just Klein group or Vierergruppe, four-group, often symbolized by the letter V or as K4) is the group, the direct product of two copies of the cyclic group of order 2.
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Laguerre's method
In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials.
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Michael Rosen (mathematician)
Michael Ira Rosen (born 7 March 1938 in Brooklyn) is an American mathematician who works on algebraic number theory, arithmetic theory of function fields, and arithmetic algebraic geometry.
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MIT Press
The MIT Press is a university press affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts (United States).
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Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician.
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Newton's method
In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
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Niels Henrik Abel
Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields.
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Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.
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Nouvelles Annales de Mathématiques
The Nouvelles Annales de Mathématiques (subtitled Journal des candidats aux écoles polytechnique et normale) was a French scientific journal in mathematics.
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Nth root
In mathematics, an nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x: where n is the degree of the root.
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Paolo Ruffini
Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician and philosopher.
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Permutation
In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.
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Pierre Wantzel
Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.
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Polynomial long division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division.
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Q.E.D.
Q.E.D. (also written QED and QED) is an initialism of the Latin phrase quod erat demonstrandum meaning "what was to be demonstrated" or "what was to be shown." Some may also use a less direct translation instead: "thus it has been demonstrated." Traditionally, the phrase is placed in its abbreviated form at the end of a mathematical proof or philosophical argument when the original proposition has been restated exactly, as the conclusion of the demonstration or completion of the proof.
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Quadratic formula
In elementary algebra, the quadratic formula is the solution of the quadratic equation.
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Quartic function
In algebra, a quartic function is a function of the form where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial.
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Quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Resolvent (Galois theory)
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root.
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Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.
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Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.
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Solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions.
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Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial splits or decomposes into linear factors.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Square root
In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.
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Symmetric function
In mathematics, a symmetric function of n variables is one whose value given n arguments is the same no matter the order of the arguments.
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Thesis
A thesis or dissertation is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.
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Topological Galois theory
In mathematics, topological Galois theory is a mathematical theory which originated from a topological proof of Abel's impossibility theorem found by V. I. Arnold and concerns the applications of some topological concepts to some problems in the field of Galois theory.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician.
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World Scientific
World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore.
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References
[1] https://en.wikipedia.org/wiki/Abel–Ruffini_theorem