Download 25. Abel`s Impossibility Theorem

25. Abel’s Impossibility Theorem
Polynomial equations of the fifth and higher degree are in general incapable of
algebraic solution (or "solution by radicals"), i.e., by addition, subtraction,
multiplication, division and roots.
This famous theorem was first stated by the Italian physician Paolo Ruffini (1765-1822)
in his book Teoria generale delle equazioni (Bologna 1798). Ruffini’s proof, however, was
not complete. The first rigorous proof was given in 1826 in the first volume of Crelle’s
Journal für Mathematik by the young Norwegian mathematician Niels Henrik Abel
(1802-1829).
Whole books have been written on this theorem, and it is often the goal in a course
(undergraduate or graduate) in Abstract Algebra, after wading through a lot of Galois
Theory. Needless to say, the proof cannot readily be condensed into several pages without
significant background. Dörrie arrives at the following theorem from Kronecker (from 1856
in the Monatsberichte der Berliner Akademie):
Theorem. An irreducible polynomial equation of odd prime degree which is solvable by
radicals has either just one real root or all real roots.
Thus, much effort has been expended to determine irreducibility criteria, and ways to tell
how many real roots polynomial equations have. For example
Eisenstein’s Criterion Let p be a prime, and fŸx a n x n . . . a 1 x a 0 be a polynomial
with integer coefficients. If p 4 a n , p|a n"1 , . . . , p|a 0 , but p 2 4 a 0 , then fŸx is irreducible
(over f).
For example fŸx x 5 " 6x " 3 with p 3, is irreducible. By Example 2 of No. 24, fŸx has three real roots and two complex roots. By Kronecker’s theorem, it is not solvable by
radicals.
Note 1. Quadratic, cubic and quartic equations can be solved by radicals.
Note 2. Graphing calculators enable the user to "zoom in" on roots of any equation.
Note 3. Newton’s method can be used to approximate roots of equations.
Note 4. The book Abel’s Proof by Peter Pesic, MIT Press, 2003, contains (a translation
of) Abel’s 1824 paper, comments about it, and about his 1826 paper that spelled out
more details of his argument.
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