History of complex numbers: Higher order polynomials

   After solving the problem of finding the roots of cubic equations and equations of 4th degree, largely due to the introduction of complex numbers, a new task naturally arose: to find a formula which, with the help of radicals, would express the roots of any algebraic equation through its coefficients. The search for such formulas began in the days of Cardano and Ferrari.

    However, although it lasted nearly three centuries, there was little success. Even for the 5th degree equation, nobody was able to find appropriate formulas. Not surprisingly, the outstanding French mathematician of the XVIII century Joseph Louis Lagrange (1736-1813) called this issue "a challenge to the human mind." And it was only in the XIX century that the cause of failure was determined. First, the Italian scientist Paolo Ruffini (1765-1822) realized and tried to prove that the problem was actually unsolvable. But his arguments were not perfect. Independently from Ruffini, the same result was reached by the outstanding Norwegian mathematician Niels Henrik Abel (1802-1829), and he fully proved it. Now the corresponding statement is called the Abel-Ruffini theorem. It is formulated in the following way: An n-th degree algebraic equation with numerical coefficients for n> 5 can not be solved by radicals.

   We should not think, however, that when the equation is not solved by radicals, it has no solutions whatsoever. Even in the XVIII century, the famous French scholar and encyclopaedist Jean d'Alembert (1717-1783) expressed, and later KF Gauss strictly proved the following theorem, which is now called the fundamental theorem of algebra: Every equation of the n-th degree (n> 1), with any numerical coefficients, has exactly n roots, complex or real, in particular (some of which, however, may be the same).

   The absence of formulas for solving algebraic equations of n-th degree for n> 5 does not create insurmountable practical difficulties for finding approximate solutions of any degree of accuracy. Interestingly, it is by approximate methods that equations of the 3rd and 4th order are solved nowadays, since these methods are much simpler than the Cardano-Ferrari methods for finding exact solutions (we provide qualitative quantitative methods help).

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