Solution By Radicals

An equation is said to be solvable by radicals if its solution can be achieved in a finite number of steps using the standard operations of addition, multiplication, subtraction, division, and extraction of roots. By this definition, all quadratic equations, cubic equations, and quartic equations are solvable by radicals. That is, there exists a formula for a general solution to any form of the equation.

Abu Ja'far Al-Kwarizmi was the first to present a systematic analysis of quadratic equations. Although he did not provide a solution to the general form (this would come from Abraham bar Hiyya Ha-Nasi in 1145), Al-Kwarizmi did show how six different categories of quadratic equations could be solved. The general solution for the quadratic equations comes from the method of completing the square. That is, transforming an equation from the form ax² + bx + c to the form (2ax + b)² = b² -4ac.

The cubic equation has an interesting history with the solution of the depressed cubic equation made by Scipione del Ferro and Nicolo Fontana Tartaglia. Later, Gerolamo Cardano showed how the general cubic equation could be reduced to the depressed cubic equation. The general solution for quartic equations was first solved by Lodovico Ferrari who was Cardano's student.

Niels Henrik Abel provided the first complete proof that quintic equations or any other equation of higher degree is not solvable by radicals. Evariste Galois was able to establish a generalize the result through the introduction of group theory.

In the next few blogs, I will show how investigation of the quintic equation led to group theory and Galois's proof.