Thursday, November 09, 2006

Biographies of Mathematicians

Tuesday, October 24, 2006


Here is an overview of the story behind Fermat's Last Theorem Blog.

1. Pierre de Fermat

Fermat's Last Theorem was first mentioned in the notes of Pierre de Fermat that was published after Fermat's death.

Fermat was making a comment on Diophantus's Arithmetica.

2. Diophantus

Diophantus is considered the father of algebra.

Diophantus wrote about the Pythagorean Triples.

This is the problem that Fermat was commenting on when he proposed his proof.

3. Fermat's Last Theorem: n=4

Fermat himself presented a proof that can be used to proof n=4.

Pythagorean Triples can be used for n=4.

4. Leonhard Euler

Euler was one of the prolific mathematicians of all time.

Euler was able to show a proof for n=3.

Euler also proposed a very innovative proof for n=3 that contained a mistake.

Euler's mistake shows the importance of unique factorization.

5. Carl Friedrich Gauss

Gauss is considered to be one of the greatest mathematicians of all time.

Gauss proposed an extension of integers which are today called Gaussian integers.

Gaussian integers can be used to demonstrate Fermat's Last Theorem for n=4.

6. Ferdinand Eisenstein

Eisenstein was a brilliant mathematician.

Eisenstein integers are named in his honor.

Eisenstein integers can be used to demonstrate Fermat's Last Theorem for n=3.

7. Sophie Germain

Sophie Germain was a very talented mathematician who struggled against the prejudices of her times.

She came up with an important proof related to Fermat's Last Theorem that extended Fermat's Little Theorem.

8. Binomial Theorem

To make progress on Fermat's Last Theorem n=5, it is necessary to review the Binomial Theorem.

In this context, it is important to take notice of the achievements of Sir Isaac Newton and Blaise Pascal.

9. Continued Fractions

Joseph Lagrange did very important work in relation to continued fractions.

Continued fractions can be used to solve Pell's Equation.

Pell's equation is needed to solve for units with regard to quadratic integers which is a generalization of Gaussian integers and Eisenstein integers.

Evariste Galois showed an alternative solution to Pell's Equation using Reduced Quadratic Equations.

10. Fermat's Last Theorem for n=5

Fermat's Last Theorem for n=5 was solved by Adrien Legendre and Johann Dirichlet.

11. Fermat's Last Theorem for n=7

Fermat's Last Theorem for n=7 was solved by Gabriel Lame.

Lame attempted a more general proof but made a mistake similar to Euler's.

12. Ernst Kummer

Kummer made a very important breakthrough in Fermat's Last Theorem.

He was attempting to generalize quadratic reciprocity when he came up with his solution.

He proposed a theory of ideal numbers as an effort to save unique factorization for cyclotomic integers.

He then was able to provide a proof of Fermat's Last Theorem for all primes that do not divide their class number.

But Kummer's approach raised the question of how you determine the class number for a given prime.

Kummer was able to solve this problem using his class formula [details to be added].

13. Euler Product Formula

To understand Kummer's solution to the class number problem, it is necessary to review the Euler Product Formula.

This takes us into the Riemann zeta function which includes the Basel Problem, the Sum of the Reciprocals of Primes, and the general probability of two integers being relatively prime.

14. Bernoulli Numbers

In addition to the Euler Product Formula, it is necessary to understand the Bernoulli numbers.

These were written about by Jacob Bernoulli who was the brother to Johann Bernoulli.

15. Kummer's Class Number Formula

Kummer's solution to determining the class number for a given prime is Kummer's class number formula. More about this in future blogs.

This formula is based on the Zeta Function for Cyclotomic Integers. I went over the first step of the derivation.

To move ahead, it is necessary to use some properties of cyclic groups so this blog will now take a detour to review group theory.

16. Solution by Radicals

Group theory begins with a study of equations that are solvable by radicals. It has been long known that the quadratic equation are solvable by radicals. Abu Ja'far Al Kwarizmi was the first to systematize and analyze quadratic equations.

17. Cubic Equation

Girolamo Cardano was the first to show the general solution of the cubic equation which was based on work done by Nicolo Fontana Tartaglia and Scipione del Ferro.

18. Complex Numbers

Rafael Bombelli invented complex numbers based on the solution to the cubic equation.

19. Quartic Equation

Lodovico Ferrari was the first to find the general solution of quartic equations.

20. François Viète

François Viète introduced coefficients and variables and made other important contributions to the algebra.

21. Van Roomen's Problem

The first great victory of François Viète was the solution of Van Roomen's problem. This problem showed the relationship of trigonometry with algebra and established Viète as perhaps the premier mathematician of his day.

22. Newton's Identities

Newton published a series of identities that played an important role in the analysis of symmetric polynomials. This would be a very important step in the path to Galois's generalization. The identities had also been discovered earlier by Albert Girard.

23. Nth Roots of Unity

The Nth roots of unity were thoroughly studied as a response to a famous mistake by Gottfried Leibniz. Roger Cotes provided formulas to answer Leibniz which were later proved by Abraham de Moivre.

24. Solution of 5th and 7th Root of Unity

De Moivre provided a famous formula which had a simpler proof by Leonhard Euler. He was able to use this trigonometric formula to show the existence of the roots of unity. De Moivre was able to solve the 5th root of unity and the 7th root of unity by radicals.

25. Solution of Eleventh Root of Unity

Alexandre-Theophile Vandermonde
was able to solve the eleventh root of unity by radicals. He published his paper at the same time as Joseph-Louis Lagrange who provided an important formula that is today known as the Lagrange Resolvent.

26. Cyclotomic Equations

At age 19, Carl Friedrich Gauss solved the seventeen root of unity by radicals and noticed that this solution indicated that it was possible to construct a regular, seventeen-sided figure using a compass and ruler.

Later, Gauss was able to show that all roots of unity are solvable by radicals and further that a regular n-sided figure is constructible by compass and ruler if and only if n is the product of distinct Fermat primes and a power of 2.

27. Insolubility of the Quintic Equation

Paolo Ruffini was the first to attempt to prove that certain quintic equations were algebraically insoluble. There was a significant gap in his proof. Still, much of Ruffini's proof holds.

Niels Abel presented the first complete proof of the insolubility of the quintic. His proof depended on work from Augustin-Louis Cauchy who had been greatly influenced by Ruffini.

Abel's proof was not accepted right away but thanks to review by mathematicians such as William Hamilton, Abel's achievemenet was recognized.

Abel's work would later be extended by Leopold Kronecker who demonstrated that only equations with only one real root or all real roots were algebraically soluble.