Appendix

Historical Development

As a study of the transmission of Greek mathematics through the Muslim world, it is fitting to attach a brief look at the evolution of Algebra to contemporary methods, culminating in the Fundamental Theorem of Algebra. Tracing the evolution of algebra through the Fundamental Theorem of Algebra shows the influence Arab mathematicians have had on modern algebra.

Leonardo of Pisa

The first European algebraic advances occurred in the "early Renaissance," in the 13th century, beginning with Leonardo of Pisa, or Fibonacci. Leonardo lived circa 1180 - 1240 CE. Though born in Pisa, Leonardo was raised in Bugia, where he learned about commerce and arithmetic. After traveling to Egypt, Syria, and Provenece, and learning methods of calculation, he concluded that the decimal positional system (the Indian numbering system) was superior to all other numbering systems (van der Waerden 36). He returned to Pisa, studied Euclid's Elements, and wrote Book of the Abacus with information about arithmetic, algebra, and geometry. It was in this book that Leonardo introduced the famous Fibonacci sequence: 1+1+2+3+5+8+13+ ..., where an+1 = an + an-1. He also wrote Flower and Book of Squares, in which he solved the cubic equation x3 + 2x2 + 10x = 20 and found a rational solution of the generalized system x2 + a = u2 ; x2 - a = v2.

Scipione del Ferro

During the Renaissance (the 15th and 16th centuries), the countries of Italy, Spain, France, and England led the resurgence of art, science, and literature. The 16th century began the age of European algebra with the solution of cubic and quartic equations. Scipione del Ferro (1456-1526 CE) solved the equation: x3 + px = qp; q greater than 0, but kept his results secret; this was common at the time, as "the owner of a method could challenge his rival to a scientific duel and set him problems solvable by the method the rival was ignorant of. Victory in such a 'tournament' brought one fame and placed one at an advantage when it came to filling a desirable position" (Bashmakova 68). Del Ferro only passed his method to his student, Fiore, who challenged Niccolo Tartaglia (ca. 1499-1557 CE) to a duel.

Niccolo Tartaglia

Niccolo Tartaglia's story is interesting. He was born into a poor family in Brescia, and his father passed away when Tartaglia was only six years old. When the French sacked Brescia in 1512, Tartaglia was wounded in the jaw and larynx; his mother treated him with home remedies, since they were too poor to consult a doctor. The name "Tartaglia" is actually a nickname which means "stammerer" (Bashmakova 68). In spite of the challenges Tartaglia faced in his youth, he gained a great knowledge of mathematics and mechanics, and wrote an impressive treatise The New Science. After many attempts, Tartaglia discovered the solution for the above equation the very night before the duel with Fiore! Tartaglia was therefore able to solve all of Fiore's problems, whereas Fiore could not solve the mechanics problems Tartaglia asked him. A few days after the duel, Tartaglia also solved the equation: x3 = px + q; p,q greater than 0 (van der Waerden 55).

Francois Viète

Francois Viète was born in Fontenay-le-Comte, and did most of his work in mathematics while at the court of kings Henry III and Henry IV in Paris. His contribution to mathematics cannot be understated. Viète attempted to create a new science combining the geometry of the "ancients" with the ease of operations in algebra. In his An Introduction to the Art of Analysis, Viète introduced the language of formulas into mathematics; in this way, he created a literal calculus. He used literal notations of both parameters and unknowns, which made it possible to write equations and identities in a general form. Viète denotes unknown magnitudes with vowels A, E, I, O, and U and known magnitudes with consonants B, C, D, etc. He adopts the symbols "+" and "-" for addition and subtraction, introduces the symbol "=" for the absolute value of the difference of two numbers (or, |B-C| in modern notation). Viète also uses the word "in" for multiplication and "applicare" for division. Further, Viète introduced the rules: (van der Waerden 64-5)

B - (C ± D) = B - C -+ D; B • (C ± D) = B • C ± B • D

shown in modern notation, as well as operations on fractions (also shown in modern notation):

B/D + Z = (B + Z • D)/(D)

Viète's next treatise, Ad logisticam speciosam notae priores, introduces some of the most important algebraic formulas, such as: (Bashmakova 79).

(A+B)n = An ± nAn-1B + ... ± Bn; n = 2,3,4,5

An + Bn = (A + B) • (An-1 - An-2B + ... ± Bn-1); n = 3,5

An - Bn = (A - B) • (An-1 + An-2B + ... + Bn-1); n = 2,3,4,5

Viète's literal calculus was later perfected by Rene Descartes, who gave the literal calculus its modern form. At the end of the 17th century, a calculus was developed for the analysis of infinitesimals; this is seen in Newton's method of fluxions and infinite series and Leibniz's differential and integral calculus. The 18th century saw the development of a calculus of partial differentials and derivatives, and the 19th century saw the creation of a calculus of logic. Today, nearly every mathematical theory has its own literal calculus; the apparatus of formulas has become an "indispensable language of mathematics," and was introduced by Diophantus and Viète (Bashmakova 80). Viète's literal calculus allowed for his analysis of determinate equations. His treatise On Perfecting Equations establishes what is now known as Viète's Theorem: (Bashmakova 87)

For xn + a1xn-1 + ... + an-1x + an = 0,; n = 2,3,4,5

To have n solutions x_1, x_2, ... x_n, the symmetric expressions result:

x1 + x2 + ... + xn = -a1

x1x2 + x1x3 + ... + xn-1xn = a2

..................................

x1x2...xn = (-1)nan

Rene Descartes

Rene Descartes (1596-1650 CE) was both a philosopher and a mathematician. His Geometry was an attempt to reduce geometry to algebra, which created analytic geometry. He transformed Viète's calculus of magnitudes in specifying that "operations on segments should be a faithful replica of the operations on rational numbers" (Bashmakova 91). Instead of regarding the product of two segments as an area, as Viète and Greek mathematicians did, Descartes showed that the product is a segment (van der Waerden 73). He introduces a unit segment (u) and defined the products of segments a and b as the segment c which was the fourth proportional to the segments u, a, and b; i.e. u:a = b:c. Descartes then made the domain of segments into a replica of the semi-field R+, establishing the isomorphism between the domain of segments and the semi-field R+ (Bashmakova 92).

Descartes also presented his properties of equations (by "equation," Descartes meant setting a polynomial equal to zero), which lead to the development of the Fundamental Theorem of Algebra: (Bashmakova 93-4)

1. If α is a root of an equation then its left side is algebraically divisible by x - α;

2. An equation can have as many positive roots as it contains changes of sign from "+" to "-"; and as many false (i.e. negative) roots as the number of times two "+" signs or two "-" signs are found in succession;

3. In every equation one can eliminate the second term by a substitution; [as in completing the square in a quadratic equation]

4. The number of roots of an equation can [sic] be equal to its degree.

Descartes also formulated assertions of cubic and quartic equations, analyzing their constructibility by ruler and compass (assuming all its roots are real). Descartes discovered the method of undetermined coefficients, which led to his (cautious) formulation of the Fundamental Theorem of Algebra (Bashmakova 94).

The Fundamental Theorem of Algebra

Descartes first formulated the Fundamental Theorem of Algebra in the following way: "Every equation can have as many distinct roots (values of the unknown quantity) as the number of dimensions of the unknown quantity in the equation" (Bashmakova 94). Girard overcame Descartes' reluctance to include complex roots, and in his New Discoveries in Algebra in 1629 wrote that the number of solutions of an algebraic equation is equal to its degree. Other 18th century mathematicians used an equivalent version of the Fundamental Theorem:

"Every polynomial fn(x) = xn + a1xn-1 + ... + an-1x + an with real coefficients can be written as a product of linear and quadratic factors with real coefficients" (Bashmakova 95)

The first proof of the Fundamental Theorem of Algebra was given by d'Alembert in 1746, but his proof was purely analytic and was not rigorous, even compared to the standards of rigor of the 18th century. Euler (1707-1783 CE) presented his proof of the Fundamental Theorem of Algebra in 1746 as well; Euler's proof differed from d'Alembert's in that Euler looked for a purely algebraic proof. Euler reduced his non-algebraic assumptions to a minimum, using the following two assumptions: (Bashmakova 95)

I. Every equation of odd degree with real coefficients has at least one real root.

II. Every equation of even degree with real coefficients and negative constant term has at least two real roots.

Euler formulated his proof by reducing the solution of an equation of degree 2km, m odd, to an equation of degree 2k-1m1, m1 odd. Euler noted that it is sufficient to consider an equation Pn(x) = 0 for n = 2k; if n ≠ 2k then it is possible to find a value of k such that 2k-1 < n < 2k and multiply the polynomial fn(x) by 2k - n factors to get a polynomial of degree 2k. Euler therefore only proves the theorem for n = 4, 8, and 16 and for the general case n = 2k (Bashmakova 95-6).

C. F. Gauss

The first mathematician who rejected Euler's formulation was C. F. Gauss (1777-1855 CE) (van der Waerden 79, 95). His doctoral dissertation, written in 1799, was devoted to the proof of the Fundamental Theorem of Algebra. Gauss gave a largely algebraic proof without assuming the existence of roots of any form in 1815 (van der Waerden 95-9). Kronecker isolated the method of Gauss in pure form with the construction of the splitting field of a polynomial without assuming the existence of the field of complex numbers in 1880-1881. This may be one of the first examples of abstract algebra.

Though Euler's viewpoint was rejected at the beginning of the 19th century, it was adopted between the 1870s and 1880s, and became the viewpoint that triumphed in algebra over the viewpoint that presupposes the construction of a field of complex numbers which is then followed by a proof of the existence of a root in the field. It is interesting to note that the Fundamental Theorem of Algebra in Euler's proof coincides with the Weierstrass-Frobenius theorem which states that "the field of real numbers and the field of complex numbers are the only linear associative and commutative algebras (without zero divisors) over the field of real numbers" (Bashmakova 100). Gauss remained a large part of 19th century mathematics, as his theory of cyclotomic equations became the model for the investigations of Abel, Galois, and other 19th-century algebraists.

Evariste Galois

Evariste Galois (1811-1832 CE) solved the problem of algebraic solutions of equations. Galois was killed in a duel on May 30, 1832; the night before the duel, knowing he may die, Galois wrote a letter to his friend Auguste Chevalier setting out his fundamental results (van der Waerden 103). These dealt with the general theory of algebraic functions and the theory of equations. A basic understanding of Galois theory is necessary for a purely algebraic proof of the Fundamental Theorem of Algebra; a brief description of Galois theory as well as an algebraic proof of the Fundamental Theorem of Algebra (using Galois Theory) can be found in my full paper.

Please see the "Paper" section for further information on the continued develpment of algebra and a proof of the Fundamental Theorem of Algebra using Galois Theory.