Galois theory

In mathematics, Galois theory is a branch of abstract algebra.

Application to classical problems

The birth of Galois theory was originally motivated by the following question, which is known as the Abel-Ruffini theorem.

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: "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)?"

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Galois theory not only provides a beautiful answer to this question, it 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.

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Galois theory also has applications to several ruler-and-compass construction problems in geometry. For example,

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: "Which regular polygons are constructible polygons?"

Related Topics:
Polygon - Constructible polygon

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: "Why is it not possible to trisect every angle?"

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: "Why is it impossible to construct a circle whose area is the same as the area of a unit square?"

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In all cases, the construction must be accomplished by straight edge and compass alone.

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