ABSTRACT
This chapter covers the beautiful topic of Galois theory, whose origins lie in the theory of equations. Although methods of solving the quadratic equation were known to the ancient mathematicians, interest in the theory of equations was sparked in the 1540s when the cubic equation was solved by Scipione del Ferro and Niccol´o Tartaglia, and a method for solving the fourth-degree equation was discovered soon after by Lodovico Ferrari. This raised the question on whether fifth-degree equations, or even higher-order equations, might have similar solutions. Mathematicians tried for centuries, but were unable to find such a formula, raising some doubts about whether such a formula could exist. An important step was taken in 1770 by Lagrange, who pointed out that the formulas for the quadratic, cubic, and quartic equations hinged on finding a combination of the roots of the polynomial that would remain unchanged whenever the roots were permutated. Finally, in 1824, Niels Abel proved that no such formula could exist. (See the Historical Diversion on page 241.) ´ Independently, Évariste Galois gave a sharper result of determining which fifth-degree polynomials, or even higher-order polynomials, could be solved in terms of square roots, cube roots, or higher-order roots. Unfortunately, Galois did not live to see himself get the credit for his work (see the Historical Diversion on page 525.)
