Galois theory

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 roots, and the four basic arithmetic operations.

Galois theory 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), and characterizing the regular polygons that are constructible (this characterization was previously given by Gauss but without the proof that the list of constructible polygons was complete; all known proofs that this characterization is complete require Galois theory).

The birth and development of Galois theory was caused by the following question, which was one of the main open mathematical questions until the beginning of 19th century: Does there exist a 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 Abel–Ruffini theorem provides a counterexample proving that there are polynomial equations for which such a formula cannot exist.

Furthermore, it provides a means of determining whether a particular equation can be solved that is both conceptually clear and easily expressed as an algorithm.

Galois' theory also gives a clear insight into questions concerning problems in compass and straightedge construction.

This solution was then rediscovered independently in 1535 by Niccolò Fontana Tartaglia, who shared it with Gerolamo Cardano, asking him to not publish it.

After the discovery of del Ferro's work, he felt that Tartaglia's method was no longer secret, and thus he published his solution in his 1545 Ars Magna.

[3] His student Lodovico Ferrari solved the quartic polynomial; his solution was also included in Ars Magna.

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.

A further step was the 1770 paper Réflexions sur la résolution algébrique des équations by the French-Italian mathematician Joseph Louis Lagrange, in his method of Lagrange resolvents, where he analyzed Cardano's and Ferrari's solution of cubics and quartics by considering them in terms of permutations of the roots, which yielded an auxiliary polynomial of lower degree, providing a unified understanding of the solutions and laying the groundwork for group theory and Galois' theory.

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 1824, thus establishing the Abel–Ruffini theorem.

While Ruffini and Abel established that the general quintic could not be solved, some particular quintics can be solved, such as x5 - 1 = 0, and the precise criterion by which a given quintic or higher polynomial could be determined to be solvable or not was given by Évariste Galois, who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots – in modern terms, its Galois group – had a certain structure – in modern terms, whether or not it was a solvable group.

[citation needed] In 1830 Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvability by radicals; Galois' paper was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients.

Galois then died in a duel in 1832, and his paper, "Mémoire sur les conditions de résolubilité des équations par radicaux", remained unpublished until 1846 when it was published by Joseph Liouville accompanied by some of his own explanations.

[4] Prior to this publication, Liouville announced Galois' result to the Academy in a speech he gave on 4 July 1843.

[5] According to Allan Clark, Galois's characterization "dramatically supersedes the work of Abel and Ruffini.

For example, in his 1846 commentary, Liouville completely missed the group-theoretic core of Galois' method.

[7] Joseph Alfred Serret who attended some of Liouville's talks, included Galois' theory in his 1866 (third edition) of his textbook Cours d'algèbre supérieure.

In Britain, Cayley failed to grasp its depth and popular British algebra textbooks did not even mention Galois' theory until well after the turn of the century.

Dedekind wrote little about Galois' theory, but lectured on it at Göttingen in 1858, showing a very good understanding.

[8] Eugen Netto's books of the 1880s, based on Jordan's Traité, made Galois theory accessible to a wider German and American audience as did Heinrich Martin Weber's 1895 algebra textbook.

This results from the theory of symmetric polynomials, which, in this case, may be replaced by formula manipulations involving the binomial theorem.

If a factor group in the composition series is cyclic of order n, and if in the corresponding field extension L/K the field K already contains a primitive nth root of unity, then it is a radical extension and the elements of L can then be expressed using the nth root of some element of K. If all the factor groups in its composition series are cyclic, the Galois group is called solvable, and all of the elements of the corresponding field can be found by repeatedly taking roots, products, and sums of elements from the base field (usually Q).

One of the great triumphs of Galois Theory was the proof that for every n > 4, there exist polynomials of degree n which are not solvable by radicals (this was proven independently, using a similar method, by Niels Henrik Abel a few years before, and is the Abel–Ruffini theorem), and a systematic way for testing whether a specific polynomial is solvable by radicals.

Choose a field K and a finite group G. Cayley's theorem says that G is (up to isomorphism) a subgroup of the symmetric group S on the elements of G. Choose indeterminates {xα}, one for each element α of G, and adjoin them to K to get the field F = K({xα}).

Various people have solved the inverse Galois problem for selected non-Abelian simple groups.

The condition imposed by Jacobson has been removed by Brantner & Waldron (2020), by giving a correspondence using notions of derived algebraic geometry.