Skip to main content

Posts

Showing posts with the label Polynomials

Galois theory for Roots of Polynomials of Higher Degree

We all know the easy way to solve the quadratic equation,           a x²+bx+c =0. W e use the quadratic formula for roots. We also have access to similar formulae for roots of cubic equations and quartic equations. The formula for roots of a cubic equation  ax 3 +bx 2 +cx+d=0 is  A similar complex looking formulae exist for roots of a quartic equation. But myster i ously we do not have any such formula for roots of 5 or higher degree polynomials. It seems as if, we can not construct the solutions of a degree 5 or higher degree polynomials just by using addition, subtraction, multiplication, division, and radicals in its coefficients.  Why so? what is so special in this 5-degree polynomial? These were questions that haunted the young Frenchman  Evariste Galois  in the 18th century. He developed a new mathematical object called a “group” that solved this issue in a surprisingly cool way. Galois being shot in a duel.  Image from...