Lecture 1
Introduction and Overview
 
Then Cardano's formulas are
$\displaystyle y_1$ $\displaystyle =$ $\displaystyle \sqrt[3]{\frac{-27}{2} + \frac{3}{2}\sqrt{-3\delta}}$  
$\displaystyle y_2$ $\displaystyle =$ $\displaystyle \sqrt[3]{\frac{-27}{2} - \frac{3}{2}\sqrt{-3\delta}}$  

We also have $ x_1+x_2+x_3=0.$ These three linear equations determine the roots $ x_1,x_2,x_3.$

In 1536 Lodovico Ferrari entered Cardano's house as a servant. Due to his extraordinary mathematical abilities he became a mathematician under Cardano's guidance. Ferrari showed that a quartic equation can be reduced to a cubic equation and therefore it can be solved by means of four arithmetic operations and extraction of square and cube roots. We will derive the formulas of Cardano and Ferrari later.

Some of the greatest mathematicians, e.g., Euler and Lagrange attempted to find similar formula for the roots of quintic equations. Lagrange gave a general method to solve equations of degree atmost four. But this method did not work for quintic equations.

Mathematicians became skeptical about existence of such formulas for equations of degree five and higher. Paolo Ruffini, born 1765 was a student of Lagrange. He published several papers(1802, 1813) about insolvability of general quintic equation. His proof was not complete. The first complete proof was given by Neils Henrik Abel (1802-1829) in 1824. Abel also proved that if the Galois group of a polynomial is commutative then the polynomial is solvable by radicals. Commutative groups are called Abelian to honour Abel for his deep work in many branches of mathematics.

Gauss made two fundamental contributions to the theory of equations. He obtained a complete solution by means of radicals of the cyclotomic equation

$\displaystyle x^n-1=0.$