Niels Henrik Abel ( 1802 1829 ), a Norwegian, and �variste Galois, ( 1811 1832 ) a Frenchman, investigated the solutions of various polynomial equations, and proved that there is no general algebraic solution to equations of degree greater than four ( Abel Ruffini theorem ).
42.
As a boy he read some of the writings of Joseph Louis Lagrange on the solution of numerical equations, and of Niels Abel concerning the impossibility of obtaining an algebraic solution for the equation of the fifth degree, was published in the " Nouvelles Annales de Math�matiques ".
43.
It would help if you could do some manipulation on the series and find that the result is a series for some other known function that provides an algebraic solution, but in this case the series is just the sine series with a few terms altered, and that doesn't help.
44.
An "'algebraic solution "'or "'solution in radicals "'is a closed form expression, and more specifically a closed-form algebraic expression, that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots ( square roots, cube roots, and other integer roots ).
45.
This procedure produces extraneous solutions, but when we have found the correct ones by numerical means we can also write down the roots of the quintic in terms of square roots, cube roots, and the Bring radical, which is therefore an algebraic solution in terms of algebraic functions of a single variable & mdash; an algebraic solution of the general quintic.
46.
This procedure produces extraneous solutions, but when we have found the correct ones by numerical means we can also write down the roots of the quintic in terms of square roots, cube roots, and the Bring radical, which is therefore an algebraic solution in terms of algebraic functions of a single variable & mdash; an algebraic solution of the general quintic.
47.
The connection with " p "-curvature is that the mod " p " condition stated is the same as saying the " p "-curvature, formed by a recurrence operation on " A ", is zero; so another way to say it is that " p "-curvature of 0 for almost all " p " implies enough algebraic solutions of the original equation.
48.
Due to complex interactions which arise from electron-electron repulsion, algebraic solutions of the Schr�dinger equation are only possible for systems with one electron such as the hydrogen atom, H 2 +, H 3 2 +, etc .; however, from these simple models arise all the familiar ?-bonding molecular orbitals stretching through the entire molecule rather than two isolated double bonds as predicted by a simple Lewis structure.
