The quartic equation re-visited for time-passengers ...

.. Welcome in the years 1736 and 1811 ..

Joseph Louis Lagrange (1736 - 1813); Italian/French Mathematician - born in Turin, died in Paris.

Lagrange was born as the eldest of 11 children, of whom only two survived to adulthood. Despite being an autodidact and lacking the opportunity to study with leading mathematicians, Lagrange is undeniably one of the most gifted mathematicians of the 18th century. At the age of 16, he became a professor of mathematics at the Artillery School in Turin. Mathematics does not appear to have been his first choice: if he had been wealthy, he likely would not have devoted himself to mathematics. (Lagrange's father lost much money in financial speculations.)

Lagrange's name is inscribed in marble for his contributions to modern mechanics, algebra, number theory, and the calculus of variations (his 1756 work is probably his most significant achievement). This list is far from complete. He is an icon - still an understatement - of what we now call "mathematical physics." The famous Swiss mathematician Leonhard Euler once attempted to persuade Lagrange to join the Berlin Academy of Sciences, but Lagrange declined. In one of his letters from 1765, he wrote: "It seems to me that Berlin would not be at all suitable for me while M. Euler is there."

Lagrange's different approach to solutions demonstrated that he sought methods distinct from Euler's, although he held Euler in the highest respect. In 1766, Euler left Berlin for Saint Petersburg, and Lagrange became the director of mathematics at the Berlin Academy from 1766 to 1787, invited by King Frederick the Great of Prussia (1712 - 1786). During 1770 and 1771, while still in Prussia, he analyzed methods for solving equations of higher degrees (3, 4, and beyond). He showed that the roots of an algebraic equation could be expressed quite simply via the roots of the so-called resolvent, an expression introduced by Lagrange. However, these results caused considerable unease within the traditional mathematical community: the resolvent of a cubic is a polynomial of degree 2, that of a quartic is cubic (so far all peachy), but the resolvent for a quintic is of degree six. These findings cast doubt on the general solvability of equations of degree higher than four - at least, using patterns known to the mathematicians of the time. In 1786, after the death of King Frederick, Lagrange accepted an invitation from Louis XVI to move to Paris. He became a member of the French Academy of Sciences. In Paris, he suffered from severe melancholy, caused by his restless life, which damaged his health. Ironically, it was the French Revolution that temporarily freed him from this gloom.

Many of Lagrange's colleagues were executed during the Revolution - most famously Antoine Laurent de Lavoisier, the chemist and aristocrat, who was guillotined in 1794. Lagrange was deeply affected by Lavoisier's death, reportedly saying: "It took only an instant to cut off that head, but France may not produce another like it in a century." He survived the most turbulent years of the Revolution, though he was not entirely uncritical; he openly dismissed the writings of the revolutionary leader Jean-Paul Marat as "empty talk." Marat, furious at this, sought revenge during the Revolution. Napoleon Bonaparte protected scientific progress in France, partly for military reasons. Alongside Pierre-Simon Laplace, Adrien-Marie Legendre, and Sylvestre François Lacroix, Lagrange shifted from the politics of the Directory to Napoleon's regime. (the Directory, in french:"Directoire" was the government system, 1795 - 1799, of the French First Republic, and was eventually overthrown by Napoleon Bonaparte (1769 - 1816))

Napoleon appointed Lagrange to his Legion of Honor and made him a count of the Empire in 1808. He often consulted Lagrange on technical and philosophical matters. On April 3, 1813, Lagrange was awarded the Grand Croix of the Ordre Impérial de la Réunion. Only a week later, he died. Later that year, he was interred in the Pantheon as a "Father of France."

Lagrange's approach for the quartic equation

So let be now σ₁,..,σ₄ the roots of x⁴ + a₁x³ + a₂x² + a₃x + a₄ = 0, we are looking for a cubic polynomial with roots ξ₁,ξ₂ and ξ₃ such that

ξ₁ = σ₁σ₂ + σ₃σ₄
ξ₂ = σ₁σ₃ + σ₂σ₄
ξ₃ = σ₁σ₄ + σ₂σ₃

Then:

ξ₁ + ξ₂ + ξ₃ = ∑_{[i < j]}σ_iσ_j = a₂

∑_{[i < j]}ξ_iξ_j = ∑_{[i < j < k]}σ_i^uσ_j^vσ_k^w = a₁a₃ - 4a₄

ξ₁ξ₂ξ₃ = σ₁σ₂σ₃σ₄(σ₁² + σ₂² + σ₃² + σ₄²) ∑_{[i < j < k]}σ_i²σ_j²σ_k² = -4a₂a₄ + a₁²a₄ + a₃²

Where: i,j,k integers ≥ 1; u, v, w are integers ∈ {1,2} such that uvw = 2.

This way we obtain the resolvent of the quartic equation: x³ - a₂x² + (a₁a₃ - 4a₄)x + (4a₂a₄ - a₁²a₄ - a₃²) = 0

In the end, we obtain 7 equations for 4 unknown valuesthe roots of the quartic equation - so our system is overdetermined. This means we need to double-check which solutions are valid.

sample: x⁴ - 4x³ + 3.5x² + x - 0.9375 = 0

/resolvent .. deal: resolvent 1 -4 3.5 1 -0.9375 --> 1 -3.5 -0.25 0.875
resolvent:{c:x%(*)x;a33:a3*a3:c 3;a11:a1*a1:c 1;e:(4*a4*a2:c 2)-a33+a11*a4:c 4;d:(a1*a3)-4*a4;:(1f),(neg a2),d,e}
/solving with Cardano t³ - 3.5t² - 0.25t + 0.875 = 0
/we find: t₁ = 3.5, t₂ = -0.5, t₃ = 0.5
/we have to check 4 triples, the first one will be:
σ₁σ₂ = 15/4
σ₁σ₃ = 3/4
σ₁σ₄ = 5/4
/this combination would give no correct results!
/the final results at the end: x₁ = 5/2, x₂ = 3/2, x₃ = 1/2, x₄ = -1/2

Évariste Galois (1811 - 1832); French Mathematician

Galois

Lagrange's results laid the groundwork for the profound conclusion by the brilliant french mathematician Evariste Galois: Galois theory proves that there exist quintic (degree 5) and higher polynomials that cannot be solved using a formula using only radicals (means basic operations like +,-,*,% and roots)..

Galois was born in Bourg-la-Reine, nowadays a suburb of southern Paris, less than 10 km from the center of Paris. In 1823, he attended the renowned Lycée Louis-le-Grand in Paris. Before that, his only teacher was his mother, but she prepared him well for his first school. It must be said that Galois's health was not always the best; for example, during the winter of 1825, he suffered from a serious earache. At Louis-le-Grand, he had the opportunity to study Lagrange's personal memoirs, especially the Réflexions sur la résolution algébrique des équations (finalized by Lagrange around 1770/1771).

Galois was not an easy character to get along with; he was very polarizing and, to a large extent, misunderstood and/or underrated in his mathematical abilities. One of the teachers who recognized Galois's true genius was Mr. Louis Paul Émile Richard (17951849), who tried to help him. Mr. Richard realized that Galois's mathematical abilities were beyond the reach of any other students at that school. Other teachers regarded Galois as quite lazy or even underperforming; he often skipped exercises and was not very successful in taking exams. At the age of 16, he believed for a short time that he had found a general solution to polynomials of degree five.

As mentioned earlier, another aspect of Galois's complex personality was his politically "revolutionary" spirit. His dream was to study at the École Polytechnique, not only because it was the most renowned institution for mathematics but also because it was a center of new political ideas and open to revolutionmuch unlike the politically conservative Lycée Louis-le-Grand. In 1828, he failed the entrance exam for the École Polytechnique; this school would have been a place where Galois felt understood. In 1829, Galois submitted his first paper to the Académie des Sciences, but for unknown reasons, Augustin-Louis Cauchy (1789 - 1857; at that time the leading mathematician in France) either lost or misplaced the paper, at least officially.

In July 1829, his father was driven to commit suicide because his reputation had been severely damaged by a devious plot. This loss was a devastating blow to Galois, and, as if that were not enough, he failed in a second attempt to gain admission to the École Polytechnique.

Unfortunately, 1832 Galois died at the age of 20 from wounds sustained after a duel - allegedly stemming from a love dispute. As a political activist, there are rumors that the duel was a trap orchestrated by government secret agents. As a fervent republican and idealist, Galois was opposed to the monarchy and actively supported political uprisings in France. He participated in the July 1830 Revolution (also called the Second Revolution or Trois Glorieuses "Three Glorious Days"), which aimed to overthrow the ultra-royalist King Charles X.

Throughout his political life, Galois was frequently arrested. Political views also divided the scientific community. The royalist Augustin-Louis Cauchy, who as mentioned above was a professor at the École Polytechnique at the time, reviewed Galois's early mathematical papers. While not entirely unsubstantiated, some claims suggest that, for political reasons, Cauchy set aside Galois's papers and refused to publish them - an act that was a disaster for Galois. To this day, the true motivation behind Cauchy's actions remains a mystery. Galois stayed in contact with the École Polytechnique until his death but his achievements remained unrecognized. It was Joseph Liouville who published in 1846 Galois' revolutionary papers (with some additions) - 14 years after Galois' death. It is a well-established fact that Galois work ultimately answered the question that had occupied many mathematicians for centuries.

Joseph Liouville (1809 - 1882); French Mathematician and engineer

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