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Application to classical problems

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. Galois' theory provides a much more complete answer to this question, by explaining why it is possible to solve some equations, including all those of degree four or lower, in the above manner, and why it is not possible for most equations of degree five or higher. 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. 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