Answer both questions: Research and report on Lagrange’s Theorem. What was the original form of the theorem and why did Lagrange work on the problem? Who is credited with proving the modern form of...

ANS 1: Report on Lagrange’s Theorem:
Lagrange's Theorem first appeared in 1770-71 in connection with the problem of solving the general polynomial of degree 5 or higher, and its relation to symmetric function. Lagrange published a landmark work on theory of equations in 1770-71. Lagrange observed that the solutions for the cubic and quadratic equations involved solving supplementary resolvent polynomials of lower degree whose coefficients were rational functions of the coefficients of the original polynomials. He found that the roots of these auxiliary equations were in fact “functions” of the roots of the original equation that took on a small number of values when the original roots were permuted in the formulas for these functions.
Several decades later, Paolo Ruffini made further progress in Lagrages’s approach to solving polynomial equations. Rufffni showed that there does not exist any function of 5 variables taking on three values or four values. Ruffini also claimed to have proved (as a consequence) that the 5th degree equa- tion (and in general the nth degree for n > 5) was not solvable. His work drew much criticism and even though he published several more versions, his proof is generally regarded as incomplete. A friendly response came from Abbati in 1802. who gave some suggestions to improve Ruffini's proof. The given function was acted on by the n! permutations. The first row gave...