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Chapter 9

History of Polynomial Equations
Polynomial Equations with Degree Higher than Four

One of the oldest and maybe for centuries the only  area of study in Algebra had been polynomial equations. The problem was the finding of formulas that could give the roots of polynomials in relation with their coefficients.

It has been found, from historical researches, that the ancient Babylonians, who created their civilization in 2000 B.C. in Mesopotamia, knew how to find the roots of 1st and 2nd degree polynomials. Also they could approximate the square roots of numbers. They formulated the problems and their solutions mostly orally.

The next big step was done by the ancient Greeks. A group of mathematicians called Pythagoreans(5th century B.C.), proved that the square roots that appeared in the study of 2nd degree equations resulted to the existence of the irrational numbers. The Babylonians had already calculated that  but they did not wonder if there is a rational number such that . The discovery of irrational numbers is due to the Pythagoreans.

The ancient Greeks were using Geometrical designs for the solving of polynomial equations of the 1st, 2nd and 3rd degree. That is Geometrical designs made with a ruler and a pair of compasses. Traces of Algebraic representation for the solving of 2nd degree equations did not exist until 100 B.C. The mathematician Diofante in 250 B.C. introduced a form of Algebraic symbolism. The arithmetic of Diofante is for Algebra of the same importance as the elements of Euclid for Geometry. The Arabians improved Algebraic Calculus but did not manage to solve equations of the 3rd degree.

In the Middle Age, European mathematicians improved the things they learned from the Arabs and introduced new symbols. During the Renaissance the development of Algebra was remarkable, like all the other branches of mathematics.

Approximately at the end of the 15th century the University of Bologna in Italy, was one of the most famous in Europe. This fame was related with the attempt of the Bolognian mathematicians to solve the 3rd and 4th degree equations.

It seems that Professor Scipio del Ferro, who died in 1526 managed to solve the equation of the 3rd degree, without ever publishing his work. Niccolo Fontana known as Tartaglia found again the solution of the 3rd degree equation. This particular project of Fontana was published in 1545 from a wide learner doctor in Milan, Hieronimo Cardano in his work Ars Magna (The Great Art). Ars Magna also includes a method for solving polynomial equations of the 4th degree, by deducing them to equations with degree 3.

Of course, after that discovery, the effort was concentrated in finding the formulas which would give the roots of equations with degree 5 or greater than 5.

In the 18th century Josheph Louis Lagrange, influenced radically the theory of equations and approximately three years later C.F. Gauss (1777-1855) based on Lagrange's conclusions proved The Fundamental Theorem of Algebra . He also proved that the equation  can be solved and the most important conclusion was that a regular polygon with n angles can be designed with a ruler and a pair of compasses if and only if n has the form
 where distinct primes of the form .

The proof of the fact that there is not a formula for solving equations with degree 5 was given by the Norwegian N.H.Abel(1802-1829) in 1824.

Of course there was still the problem of finding the conditions that such an equation must satisfy in order to be solved. Abel was working on this problem until he died in the age of 27.

Eventually this problem was solved by the young French mathematician Evariste Galois(1811-1832). His theory virtually contains the solution of this problem. Galois wrote his conclusions in an illegible manuscript 31 pages long, the night before he died in the age of 20. The next day he was killed in a duel for a woman he barely knew. This manuscript became well known when Joseph Liouville presented it in the French Academy in 1843.

With the theory of equations the famous unsolved problems of ancient times were solved:

The Platonic Philosophy presumed that the straight line and the circle were the perfect lines. Therefore for the accomplishment of the above designs the ruler and a pair of compasses should be sufficient.

The proofs for the impossible of these particular designs are based on the theory of equations. The proof for the impossible of the squaring of the circle with a ruler and a pair of compasses was found later from the other two. This was when the German mathematician F.Lindemann(1852-1939) proved that the number  could not be designed with a ruler and a pair of compasses.