On the second memoir of the last letter of Évariste Galois

Klein, Kronecker, and Hermite are usually credited with solving the general quintic equation, although the decisive (and unique) construction which aids to establish a connection between the roots of the original equation and the corresponding torsion points of the corresponding elliptic curve is indicated by Évariste Galois in the second memoir of his last letter (consisting of three memoirs on seven pages). In his first memoir, Galois already surpassed the outstanding contributions of Gauss, Ruffini and Abel (which became the basis of the theorem on the unsolvability of the general quintic via radicals) with a completely unexpected and profoundly new formulation of a simultaneously necessary and sufficient criterion for the solvability of any algebraic equation via radicals! His, rightfully, completely revolutionary approach became the foundation of the so-called Galois theory, which marked the advent of a new era in mathematics. Yet, surprisingly, Melvin Kiernan's opinion that "the second and third memoirs did not influence subsequent mathematics" reflects non ceasing attempts at standard interpretations of Galois theory (based solely on the first memoir). And by the title of the book: "The development of Galois Theory from Lagrange to Artin" one easily guesses that its author sought to instill in the reader that the role of Galois is comparable to the outstanding role of many others. The concept of the irreducibility of a polynomial (from the first memoir) has become an integral part of the modern widespread presentation of Galois theory, however, the concept (from the second memoir) of depressing the degree of a polynomial (albeit irreducible), which has become the key to solving the general equation of the fifth degree, has not received any nearly comparable acknowledgement. And nowadays, as the dogma, concerning the non-constructiveness of the Galois theory continues to rage, we set the goal of our report to be no less than the complete liberation from the primitive interpretation of the (profoundly constructive) Galois theory and the recognition of its significance in the development of highly efficient algorithms for calculating exact (never to be confused with merely "approximate") solutions of applied (technical) problems!
Article “On the second memoir of the last letter of Évariste Galois” http://cte.eltech.ru/ojs/index.php/kio/article/view/1544
Homepage https://semjonadlaj.com