Galois theory

In mathematics, Galois theory provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood. It has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cube and trisecting the angle; a third problem of antiquity, squaring the circle, is also unsolvable, but this is shown by other methods); showing that there is no quintic formula; and showing which polygons are constructible.Why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?... the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation. In mathematics, Galois theory provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood. It has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cube and trisecting the angle; a third problem of antiquity, squaring the circle, is also unsolvable, but this is shown by other methods); showing that there is no quintic formula; and showing which polygons are constructible. The subject is named after Évariste Galois, who introduced it for studying the roots of a polynomial and characterizing the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is solvable by radicals if its roots may be expressed by a formula involving only integers, nth roots and the four basic arithmetic operations. The theory has been popularized among mathematicians and developed by Richard Dedekind, Leopold Kronecker, Emil Artin, and others who in particular interpreted the permutation group of the roots as the automorphism group of a field extension. Galois theory has been generalized to Galois connections and Grothendieck's Galois theory. The birth and development of Galois theory was caused by the following question, whose answer is known as the Abel–Ruffini theorem: Galois' theory not only provides a beautiful answer to this question, but also explains in detail why it is possible to solve equations of degree four or lower in the above manner, and why their solutions take the form that they do. Further, it gives a conceptually clear, and often practical, means of telling when some particular equation of higher degree can be solved in that manner. Galois' theory also gives a clear insight into questions concerning problems in compass and straightedge construction. It gives an elegant characterization of the ratios of lengths that can be constructed with this method. Using this, it becomes relatively easy to answer such classical problems of geometry as Galois' theory originated in the study of symmetric functions – the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots. For instance, (x – a)(x – b) = x2 – (a + b)x + ab, where 1, a + b and ab are the elementary polynomials of degree 0, 1 and 2 in two variables. This was first formalized by the 16th-century French mathematician François Viète, in Viète's formulas, for the case of positive real roots. In the opinion of the 18th-century British mathematician Charles Hutton, the expression of coefficients of a polynomial in terms of the roots (not only for positive roots) was first understood by the 17th-century French mathematician Albert Girard; Hutton writes: