so really it's nothing to raise an eyebrow about
Ethereal said:A quick question:
Does anyone know of any good web sites which contain a large collection of mathematical proofs? It's a little frustrating having to search the Web with Google only to find nothing.
Evariste Galois (1811-1832), a French mathematician, proved that it is not possible to solve a 'general' fifth (or higher) degree polynomial equasion by radicals. Of course, we can solve particular fifth-degree equations such as x^5 - 1 = 0, but Abel and Galois were able to show that no general 'radical' solution exists.
Evariste Galois was 21 years old when he died.Galois's Theorem:
An algebraic equation is algebraically solvable if its group is solvable. In order that an irreducible equation of prime degree be solvable by radicals, it is necessary and sufficient that all its roots be rational functions of two roots.
Abel's Impossibility Theorem:
In general, polynomial equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions. This was also shown by Ruffini in 1813 (Wells 1986, p. 59).
...states that there is no general solution in radicals to polynomial equations of degree five or higher.
Reference:In the modern analysis, the reason that second, third and fourth degree polynomial equations can always be solved by radicals while higher degree equations cannot is nothing but the algebraic fact that the symmetric groups S2, S3 and S4 are solvable groups, while Sn is not solvable for n=>5.