In the opinion of the 18th-century British mathematician Charles Hutton , [2] 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 He was the first who discovered the rules for summing the powers of the roots of any equation. In this vein, the discriminant is a symmetric function in the roots that reflects properties of the roots — it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct complex conjugate roots. See Discriminant:Nature of the roots for details. The cubic was first partly solved by the 15—16th-century Italian mathematician Scipione del Ferro , who did not however publish his results; this method, though, only solved one type of cubic equation. In this book, however, Cardano did not provide a "general formula" for the solution of a cubic equation, as he had neither complex numbers at his disposal, nor the algebraic notation to be able to describe a general cubic equation.

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In the opinion of the 18th-century British mathematician Charles Hutton , [2] 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 He was the first who discovered the rules for summing the powers of the roots of any equation.

In this vein, the discriminant is a symmetric function in the roots that reflects properties of the roots — it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct complex conjugate roots. See Discriminant:Nature of the roots for details.

The cubic was first partly solved by the 15—16th-century Italian mathematician Scipione del Ferro , who did not however publish his results; this method, though, only solved one type of cubic equation. In this book, however, Cardano did not provide a "general formula" for the solution of a cubic equation, as he had neither complex numbers at his disposal, nor the algebraic notation to be able to describe a general cubic equation.

With the benefit of modern notation and complex numbers, the formulae in this book do work in the general case, but Cardano did not know this.

It was Rafael Bombelli who managed to understand how to work with complex numbers in order to solve all forms of cubic equation. Crucially, however, he did not consider composition of permutations. The quintic was almost proven to have no general solutions by radicals by Paolo Ruffini in , whose key insight was to use permutation groups , not just a single permutation.

His solution contained a gap, which Cauchy considered minor, though this was not patched until the work of the Norwegian mathematician Niels Henrik Abel , who published a proof in , thus establishing the Abel—Ruffini theorem. This group was always solvable for polynomials of degree four or less, but not always so for polynomials of degree five and greater, which explains why there is no general solution in higher degrees.

Originally, the theory had been developed for algebraic equations whose coefficients are rational numbers. It extends naturally to equations with coefficients in any field , but this will not be considered in the simple examples below. These permutations together form a permutation group , also called the Galois group of the polynomial, which is explicitly described in the following examples. First example: a quadratic equation[ edit ].

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## Galois theory

Emil Artin Notes by Albert A. Published under the title Modern Higher Algebra. This volume became one of the most popular in the series of lecture notes published by Courant. Many instructors used the book as a textbook, and it was popular among students as a supplementary text as well as a primary textbook. Because of its popularity, Courant has republished the volume under the new title Algebra with Galois Theory. Readership Undergraduate students interested in algebra.

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## Algebra with Galois Theory

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## Algebra with Galois Theory

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## Emil Artin

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