Abstract:
The talk is devoted to Ostrogradsky's view on the problem of solving algebraic equations in radicals for degrees higher than four.
The proof of the impossibility of solving such equations is presented in his course “Algebraic and Transcendental Analysis”, delivered at the Naval Cadet Corps in 1837. The proof is based on Abel's work published in 1826 in Crelle's Journal. Ostrogradsky substantiates and proves all statements concerning the extension of the coefficient field of a given equation—points that Abel had only formulated. To explain the operations that constitute the structure of the field and the place of the equation's root within this structure, Ostrogradsky introduces a new symbol, $\nabla$ (nabla), to denote the root. The report shows the purpose and objectives of this innovation. Of great interest are the introduction of the concepts of symmetric and similar functions into the course, as well as the laws of permutation of elements in functions and the effect of such permutations on the value of the function.
Keywords: algebraic functions, roots of equations, symmetric functions, permutations.
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