Here is an overview of the story behind
Fermat's Last Theorem Blog.
1. Pierre de FermatFermat'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. DiophantusDiophantus 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=4Fermat himself
presented a proof that can be used to proof
n=4.
Pythagorean Triples can be used for
n=4.
4. Leonhard EulerEuler 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 GaussGauss 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 EisensteinEisenstein 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 GermainSophie 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 TheoremTo 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 FractionsJoseph 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=5Fermat's Last Theorem for n=5 was solved by
Adrien Legendre and
Johann Dirichlet.
11. Fermat's Last Theorem for n=7Fermat'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 KummerKummer 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 FormulaTo 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 NumbersIn 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 FormulaKummer'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 RadicalsGroup 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 EquationGirolamo 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 NumbersRafael Bombelli invented
complex numbers based on the solution to the cubic equation.
19. Quartic EquationLodovico Ferrari was the first to find the
general solution of quartic equations.
20. François VièteFrançois Viète introduced coefficients and variables and made other important contributions to the algebra.
21. Van Roomen's ProblemThe 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 IdentitiesNewton 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 UnityThe
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 UnityDe 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 EquationsAt 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 EquationPaolo 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.
