Galois Theory: Fifth-Degree Polynomials and Some Ruler-and-Compass Problems

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2020-05

Authors

Loehr, Gabriel Joseph

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Abstract

Évariste Galois was a French mathematician in the beginning of the 19th century. Unfortunately, his story is a tragic one. Unappreciated by his contemporaries, he struggled to gain recognition for his work and was denied entry into the top Parisian University to study math. Relegated to a second-tier school and feeling like he would never succeed, he lost his life in a duel at the age of 20. However, the night before his duel, he scribbled notes furiously and sketched out a solution to a central problem in mathematics. It would take more than a decade for his work to come to light, but he had actually managed to prove the non-existence of a general radical solution for fifth-order polynomial equations, a problem that had plagued mathematicians for centuries. Perhaps the most beautiful part of his work was its success in demonstrating the unity of different fields of math. His solution answered three seemingly unrelated problems in geometry that had remained unsolved since the time of the ancient Greeks: the duplication of the cube, trisection of the angle, and quadrature of the circle, which asked respectively for a cube whose volume is twice the volume of a given cube, an angle one-third the size of a given angle, and a square of area equal to the area of a given circle. These three problems, like fifth-order polynomials, were incredibly simple to formulate but impossible to solve. Galois’ work that settled these questions for good. In this paper, I sketch the outline of Galois’ work and show how it led to these unexpected results.

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