Abel and the Insolvability of the Quintic: Part 1



Most of the students come across the solution of linear and quadratic equations in their secondary classes. While the solution of a linear equation $ax + b = 0$ with $a, b$ being rational does not present any difficulties (because the solution $x$ itself turns out to be a rational number), a quadratic equation of the form $ax^{2} + bx + c = 0$ (with $a, b, c$ rational) does present significant challenges. For one thing the solution may not be rational and sometimes may not be even real. Usually one encounters the use of square roots to solve such an equation. Fortunately there is a standard formula for solving such equations $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$ so that the equation can be solved directly in terms of its literal coefficients.

Many mathematicians tried to extend these ideas to solve the equations of third and fourth degrees. Thus during the 16th century Cardano solved the cubic and Ferrari solved the quartic equation. Later in the 18th century Lagrange published his classic work "Reflexions sur la resolution alg├ębrique des equations" in which he unified the existing methods of solving equations upto degree $4$. He hoped that unifying all the available approaches into one coherent theory would help in solving higher degree equations. But neither Lagrange nor any other mathematician was able to provide a solution to quintics (equations of degree $5$) or higher degree equations. Then in 1824 a young Nowergian mathematician Niels Henrik Abel proved that it is not possible to solve a quintic equation in the same way as it is possible to solve equations of degree $2, 3$ or $4$.