Evariste GaloisVimal Bellamkonda

Portrait of Evariste Galois

Galois’ life

The French mathematician Evariste Galois had a very interesting life, in a time and place of great revolution.He was born on October 25th of 1811 in a small commune name Bourg-La-Reine, about 10 kilometers southof Paris. The commune is now a suburb of Paris. His family was well educated and respected. Both ofGalois' parents were educated in philosophy, religion, and classical literature. These were the subjects whichwere considered to be the most important at the time. Interestingly, neither parents had any substantialbackground in mathematics. Neither, until a certain age, did Galois himself. One of the quirks that his father,Nicholas-Gabriel Galois, had was his talent for writing rhymed couplets. Evariste Galois supposedly inheritedthis talent, and he consistently entertained his family at gatherings.

When Galois was only ten years old, his parents decided to send him to college in Reims. This would have beena great opportunity to kickstart his career in mathematics, but at the last moment, his mother unenrolled himand decided to keep him at home. As a result, Galois continued to be educated in Latin, Greek, and rhetoric,but his background in mathematics didn't start until later.

When Galois was twelve, he enrolled in his first school, the lycee of Louis-le-Grand, located in Paris. Evi-dently, however, he didn't like it. He described the school as seeming more like a prison. Additionally, theschool underwent massive political changes, involving multiple rapid successions of headmasters, leaving it inadministrative turmoil. Some sort of political student rebellion also took place, leading to the expulsion of40 students. Nevertheless, Galois had no involvement in the rebellion, and made it through his first year,receiving several awards.

When enrolled in school, Galois actually enrolled as a third year student, skipping the first two classes. This,coupled with his success in school, are evidence that the preparation being home schooled helped him greatly.However, during the school year of 1825 to 1826, his attitude and performance in school changed greatly. That

winter, he underwent serious earache, most probably due to dirty conditions in his school building. This causeda drop in his performance that eventually led to the headmaster of the school writing a letter to Galois' father,advising that Galois repeat the year. However, after considerable pushing from his father, Galois was able toresume class as normal.

The Uniform for the Louis-le-Grand at the time that Galois attended

February of 1827 was a turning point for Galois, as that was when he started his first course in mathematics,by M. Vernier. In this class, he worked with Legendre's text on geometry and become familiar with the theoryof equations outlined in Lagrange's works. Galois lagged behind in his other courses, instead pouring hisattention into math. One teacher wrote of him:

“What's dominating him is the fury of mathematics; also I think that it would be better for him if his parentswould agree to let him study solely mathematics. He is wasting his time here and he does nothing but tormenthis teachers and by doing so heaps punishments on himself. ”

Galois seemed to have a natural gift for mathematics. His teachers described him as having an extremely quickrate of learning. However, he supposedly paid little attention to homework, either making trivial mistakes ornot doing it altogether.

Galois made it his mission to attend the Ecole, as this was the single best school to launch his career inmathematics. As of ten years prior, the Ecole no longer officially were militarily oriented. Still, though, theirmission was to train scientists for government service, making it a very politically charged atmosphere. In Juneof 1828, Galois took the entrance exam at Ecole. In a surprising turn of events, he actually failed the examand didn't make it in. Nevertheless, he published his first mathematical paper in April 1829, on continuedfractions in the Annales de mathematiques. On May 25th and then again on June 1st, he published articleson the algebraic solution of equations in the journal, Academie des Sciences.

In July 1829, tragedy struck his life. That year, tensions were rising between political and religious factions inFrance. These tensions extended to Bourg-la-Reine, where a young priest, seeking the downfall of the mayorNicolas-Gabriel Galois, banded together with more of his opponents and spread vulgar poems, with Evariste'sfather appearing as the author. This was a huge scandal, and he was forced to leave Bourg-la-Reine and flee

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to Paris. Once there, he rented a small apartment, where he committed suicide. His funeral turned into aprocession of liberals, and eventually the priest was brought to justice.

Evariste Galois, despite his devastation over his father's death, decided to take the entrance exam to the Ecoleonce again. This time, however, the stakes were higher. If Galois failed, he would not be allowed to take thetest again. According to the story, Galois faced two examiners, one of whom, asked him to describe the theoryof the arithmetic logarithms. Galois, with all of his mathematical genius, criticized his the professor, statingthat there are no arithmetic logarithms. He asked why the professor had not instead asked for the theory ofthe logarithms, refusing to explain the obvious (according to him) propositions about logarithms.

One account even depicts Galois unable to find a use for the chalk and sponge with which he was supposedto write his explanations. Upon being asked the offensive, he threw the sponge at the professor's face. Whileit is likely that this is a heavily dramatized account of events, it still accurately portrays Galois' struggle tocommunicate the ideas that he held in his head. While it is indisputable that he had a genius level intellectand brilliant ideas, he consistently failed to effectively communicate those ideas to others.

At one point earlier in his schooling, Galois, seeking to be involved in the political uprisings that surroundedhim, joined the “Societe des Amis du Peuple, ” French for “Society of Friends of the People. ” It comprisedthe most active republicans who were portrayed as a very dangerous, subversive group, particularly by thegovernment-controlled press.

Galois gained a reputation for being radical and dangerous among his peers. He asked for the students to bearmed and wear uniforms in order to help train them for the military. His ideas frequently brought him intoconflict with authority.

On December 9th, Galois was publicly expelled from the school. It was believed that he had written ananonymous letter to the school paper which framed the chancellor for the events of the revolution earlier thatyear, despite him never admitting to writing the letter. Galois enlisted in the Artillery of the National Guard,which was comprised mostly of republicans, immediately after.

However, on December 31st, 1830, the National Guard was disbanded, and Galois found himself without work.With his mother unable to support him financially, he attempted to start a private class in algebra to supporthimself. On the following January 13th, it started with about 40 kids attending. However, the students wereunable to follow Galois' abstract lectures.

He continued to be involved in radical events, leading to his arrest and trial, with his ultimate acquittal.However, shortly after, he was arrested for dressing in the forbidden Artillery Guard uniform and for carryingweapons. He was sentenced to nine months in prison until April 1832.

His stay there was cut short by the Cholera epidemic that took hold of the world in 1832. Several prisoners,including Galois, were transferred to a clinic for their safety. While in this clinic, he met Stephanie, one of thedaughters of the doctors. While the nature of their relationship is not very clear, it can be gleaned from theirletters that Galois was irrevocably in love with Stephanie, but ultimately the feeling was not reciprocated.

Perhaps the most dramatized and most interesting part of Galois' life is the very endthe duel that killed him.While there have been several theories as to how the duel came about, it is most likely that Galois was simplyweary of life, due to his rejection by Stephanie, his lack of recognition for his work, his persecution for hispolitical activism, and his terrible financial situation. Thus, the duel was more of a suicide for him.

Contrary to what many believe, Galois did not write out his mathematical theory the night before he died.However, he did compose a letter to his friend Chevalier that outlined the scientific progress he had made, sothat all his work would not be lost.

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Galois’ mathematical works

The last lines of Galois' letter to Chevalier, written May 29, 1832, before his death, are as follows:

“Tu prieras publiquement Jacobi ou Gauss de donner leur avis, non sur la vrit, mais sur l'importance desthormes.

Aprs cela, il y aura, j'espre, des gens qui trouveront leur profit dchiffrer tout ce gchis. ”

This translates to, “Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to theimportance of these theorems. Later there will be, I hope, some people who will find it to their advantage todecipher all this mess. ” Jacobi and Gauss were, of course, to very prominent mathematicians of the time.This seemed to be Galois's last hope that his life's work could make some sizeable contribution to the field ofmathematics.

In reality, Galois's “works” were actually just a collection of notes that only amounted to about 60 pages.However, contained within those 60 pages were truly groundbreaking ideas which have had great impacton many branches of mathematics. His work had much overlap with that of Niels Henrik Abel, one of hiscontemporaries.

Galois was the first to use the word “Group” in the sense that it is used today. This cemented his place as oneof the founders of group theory, a branch of algebra. He said that if, in the decomposition of a group into itsleft and right cosets, the left and right cosets coincide, then the decomposition is a “proper decomposition.”Today, this is known as a “normal subgroup.” Additionally, he theorized the “finite field,” which today isknown as a “Galois field.”

He then went on to make basic studies concerning linear groups over finite fields, which are as follows:

• He studied the Galois group (another one of his contributions to be later explained) of the generalequation pv. In doing so he constructed the general linear group over a prime field, GL(v, p).

• He constructed a second family of finite simple groups, (after alternating groups) the projective speciallinear group PSL(2, p). He constructed them as fractional linear transforms, and he discovered that aslong as p is not 2 or 3, then they are simple.

• However, there is an exception to the above rule. PSL(2, p) is simple and acts only on p points if andonly if p is 5, 7, or 11.

However, the contribution for which he is most well-known is known as the Galois Theory. He learned that thealgebraic solution to a polynomial equation has a certain relationship with the structure of the Galois group,or a group of permutations associated with the roots, of the polynomial. Additionally, if subgroups of theGalois group can be found, and each one is normal or the Galois group is solvable, then the equation can besolved with radicals. This finding was adopted my later mathematicians, who applied it to other fields afterGalois's time. His manuscripts were published in 1846 in the Journal of Pure and Applied Mathematics. In1870, group theory was fully established in Camille Jordans Treaty Substitutions.

One approach to Galois's theory is the permutation approach. With a given polynomial, it is possible thatsome of the roots are connected by algebraic equations. For example, given two roots, X and Y , it could bethat

2X + 4Y = 32

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The idea behind this approach is to consider each permutation, or rearrangement, of the roots for which thealgebraic solution is still true. All of these permutations comprise a permutation group, or the “Galois group”of the polynomial. Note that this is being restricted to rational numbers, although the coefficients could bespecified to lie in a certain other field. Now, let us consider two examples. In the first, we look at a simplequadratic equation:

x2 − 4x+ 1 = 0

Application of the quadratic formula provides us with the two roots

A = 2 +√

3

B = 2−√

3

Some algebraic equations that hold true for the roots are

A+B = 4

AB = 1

In any of these algebraic equations with rational coefficients, swapping the two roots (in this case A and B)gives another true equation.

The Galois group of the polynomial x2 − 4x + 1 = 0, therefore, has two permutations. The first is theidentity permutation, in which the two roots are untouched (the first equation of the two). The second is thetransposition permutation that exchanges the two roots, A and B. Ultimately, it can be seen that given anyquadratic polynomial of the general form:

x2 + bx+ c = 0

There are two possible cases. Either the polynomial will have rational roots. For example:

x2 − 8x+ 12 = 0

(x− 6)(x− 2) = 0

In this case, the Galois group only contains the one permutation, the identity permutation. Alternatively, ifthe polynomial has two irrational roots. For example

x2 − 8 = 0

The Galois group contains two permutations. This is the case in the first example above.

In the second example, consider the polynomial

x4 − 10x2 + 1

This can be rewritten as(x2 − 5)2 − 24

This polynomial has four rootsA =

√2 +√

3

B =√

2−√

3

C = −√

2 +√

3

5

D = −√

2−√

3

Since there are four roots, there are 24 (or 4!) combinations of them. However, we are only interested inmembers of the Galois group, which preserve any algebraic equation with the rational coefficients A, B, C,and D. Thus, the equations that satisfy this requirement are

AB = −1

AC = 1

A+D = 0

If φ is a permutation in the Galois group, then we have:

φ(B) =−1

φ(A)

φ(C) =1

φ(A)

φ(D) = −φ(A)

As a result, the image of A can be used to define the permutation, and the Galois group has 4 elements, whichare as follows

(A,B,C,D)→ (A,B,C,D)

(A,B,C,D)→ (B,A,C,D)

(A,B,C,D)→ (C,D,A,B)

(A,B,C,D)→ (D,C,B,A)

The Galois group is therefore isomorphic to the Klein four-group.

Collaboration with other scholars

Galois did not directly collaborate with any other notable scholars throughout his life. (If he did, there is nohistorical record of it) However, he extensively studied the work of two of his contemporaries. The first of thetwo was Adrien-Marie Legendre for his work on geometry. The second was Joseph-Louis Lagrange for his workon algebra.

In addition, in his work on looking at roots of an equation, Galois worked, not in tandem with, but parallel toNiels Abel. In 1770, Lagrange first decided to treat the roots of an equation as their own objects and studiedtheir permutations. In 1799, Paolo Ruffini, an Italian mathematician, tried to solve a general quantic equationusing radicals. Both of these were steps toward the Galois theory. However, unknown to Galois, Niels Abelgave a correct proof of Ruffinis work in 1824. Interestingly, there is no record of any correspondence betweenAbel and Galois.

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Historical events that marked Galois’ life.

Galois was born towards the end of the First French Empire under Napoleon. Around the time of, andshortly preceding, his birth, unrest had been brewing in the nation. Napoleon had just finished putting downa revolt in Germany when the Tsar of Russia organized an uprising against him. Napoleon had no choicebut to advance against Russie, in order to maintain his access to the Mediterranean. His advance had minorvictories, including the taking of Smolensk, the Battle of Borodino, and his entry into Moscow. However, theharsh climate of the country proved too detrimental to his troops, forcing an arduous retreat back throughRussian soil during the winter. Meanwhile, the rest of Europe had the Napoleonic empire in their crosshairs.He was continually pushed back and forced to retreat to the frontiers of 1809. He then refused a peace offeringextended by Austria at the Congress of Prague. He did so from a fear of losing control of Italy.

Napoleon was then forced to retreat from Germany. He lost Spain to an allied army led by Wellington. Aftermultiple continued defeats, on March 30th 1814, Paris capitulated. On April 11th, 1814, Napoleon abdicatedthe throne, signaling the end of the empire. Galois, by this time, had just been born. Napoleon escapedthe country with a sparsely manned ship. The troops sent to arrest him ended up joining his side, helpingNapoleon in regaining the throne, albeit unsuccessfully. He returned to the throne in 1815 and revived hisempire. This short-lived kingdom became known as the Hundred Days. However, at the Battle of Waterloo,he was defeated and he surrendered. He was then exiled to a remote island, Saint Helena. There, he liveduntil his death in 1821. After this, the Bourbon monarchy was restored and Louis XVIII ascended the throne.

A depiction of the Battle of Waterloo by early 19th Century Irish painter William Sadler II

Thus began the Bourbon Restoration, a period of great conservative rule, which Galois, as a young liberal,would come to have many problems with. It lasted from 1814 until 1830, almost directly aligning with thelife of Galois. The former foreign minister of Napoleon, Talleyrand, gave considerable support to Louis XVIII,allowing the latter to ascend the throne in 1814. Formerly, the allied nations were in disagreement about whoshould take the throne. Britain wanted the Bourbons, Astria favored Francois Bonaparte, the son of Napoleon,the Russians were in favor of Louis Philippe or Jean-Baptiste Bernadotte, the former Marshal of Napoleon.Napoleon himself was offered the throne, contingent on him pulling back the frontier of France. However, herefused the offer, giving way to new authority.

Louis approved the Charter of 1814, a new constitution. It established a bicameral legislature headed by a

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Chamber of Peers (appointed) and a Chamber of Deputies (elected). However, the King held the ultimateauthority to propose or approve laws and appoint ministers. Only 1 percent of the population was eligible tovote. Many reforms from the Napoleonic era were left untouched, not just legal, but also administrative andeconomic. This included the Napoleonic Code, which had established certain civil rights and legal equality,made the biens nationaux, and divided the country into specific departments.

Louis made a large effort to display his intention of undoing the results of the French Revolution. This,however, brought dissent and lost him support from the majority. These groups included the army, any non-Catholics, and workers who had taken an economic hit by the post-war slump or from British imports. As aresult, discontent began to settle into large groups.

Napoleon was informed of this discontent and sought to capitalize on it by reclaiming his throne. On March20th of 1815, he made his way to Paris. Ironically, the large majority of the troops deployed to halt his progressended up joining him. This even included the troops that tended to be more royalist. Thus, on March 19th,Louis was forced to flee Paris. Of course, Napoleon was once again defeated at the Battle of Waterloo, and hewas exiled, thus ending the period known as the Hundred Days.

Then came the period known as the Second Restoration, in 1815. Once again, Talleyrand played an influentialrole in reinstating the Bourbons into power. Initially, the Second White Terror, during which any people whowere thought to be associated with the former Napoleonic government were arrested and then executed, began.Largely in the southern region, extreme supporters of the monarchy set out to get revenge on the supportersof the old Napoleonic government. The attackers came to be known as Verdets, due to the green color of theircockets. After a certain time, it became obvious that the local authorities were woefully unable to prevent theattacks, and the personal officials of the King and the ministers were sent to quell the violence.

On November 20th, 1815, the Second Treaty of Paris was signed. In this treaty, France was forced to pay 700million frances, and the borders of the countries were retracted to the point they were at in 1790. After theBattle of Waterloo, 1.2 million foreign soldiers were stationed in France. This occupation persisted until 1818,the most persistent of which were the 200,000 soldiers of the Duke of Wellington. As further punishment,France was made to pay the costs of their accommodations. The exorbitant costs of these reparations madethe promised tax cuts prohibitively expensive. All of these factors strengthened opposition against Louis.

A map showing the South-east frontier of France after the Second Treaty of Paris

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While the government consisted initially of moderate ministers, their policies were eventually thrown out bythe royalist majority. This majority shifted the law towards punishing enemies of the state. Over 50,000 civilservice members were sacked and 15,000 army officers were dismissed. The newly appointed prime minister,Richelieu, actually had very few ties with France. The place of the monarchy and church were aggressivelyupheld, and the ultra-royalists continued to implant their politics into the state laws.

In September of 1816, Louis dissolved the chamber, resulting in a newly formed, more liberal chamber. OnDecember 29, 1818, Richelieu ended his term as prime minister, giving the position to Jean-Joseph, MarquisDessolles, who served until November 19th, 1819.

By 1820, the liberals, who now made up half of the chamber, had grown so considerably in power that the kingand the current prime minister, Decazes, sought new ways to revise laws and institute another conservativemajority. However, in 1820, the assassination of the Duc de Berry by a Bonapartist caused Decazes to fallfrom power, and the Ultras ultimately won. For a short time after, until 1821, Richelieu returned to his seat.Afterwards, however, the Ultras found themselves back in power. Their support only continued to grow forthe next few years.

On September 16th, 1824, Louis XVIII died and was succeeded by his brother, Charles X. However, it is moreimportant to consider large political movements during this era than just the change of a single leader. From1827 to 1830, the conservative Bourbons finally fell. The specific reasons for the downfall are not completelyclear, but a few causes stand out. First of all, during this three year period, France faced a huge economicdownturn. Bad harvests put pressure on farmers, pushing them to lobby for relaxation of tariffs, in order tohelp their economic troubles. King Charles, however, kept them in place, favoring the wealthier landowners.Paris saw a decrease in overall economic activity, raising poverty levels. Politically, a large group of liberalsmanaged to make their way into the Chamber of Deputies. This group fought for more liberal economic policiesand the right to appoint the Prime Minister. Moreover, an increasingly liberal press in France ideas that rancontrary to those of the conservative government. This may have been the single most influential change, as iteducated people on different ideas than those that the government had forced on them for so long. Ultimately,the growing liberal opposition resulted in the end of the conservative power in France.

Significant historical events around the world during Galois’ life

Some of the most profoundly impactful world events during Galois' life were without a doubt in France.However, numerous other nations around the world saw significant advancements. In 1819, Simon Bolivarliberated New Granada, (which comprised what are now known as Colombia, Venezuela, and Ecuador) andSpain began to lose its control on South American countries. In 1821 Spain lost this hold entirely, andGuatemala, Panama, and Santo Domingo declared their independence.

1820 saw the Missouri Compromise in the United States, in which Missouri was admitted to the union as aslave state, but the rest of the land in the Louisiana Purchase prohibited slavery.

In 1822, the Greeks declared themselves a republic, independent from Turkey. At this time, the Turks invadedGreece, the latter of whom was aided by France and Britain. In 1828, Russia declared war on Turkey. In 1829,the war ended and the Turks recognized the Greek independence. Also in 1822, Brazil became independent ofPortugal.

In 1824, Mexico became a republic three years after declaring independence from Spain. Bolivar went on toliberate Peru and become its president. Beethovens Ninth Symphony was also composed in this year.

1825 saw the first passenger-carrying railroad in England. In 1826, Joseph-Nicephore Niepce took the firstphotograph in history.

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In 1831, the Polish revolt against Russia failed. Belgium separated from the Netherlands, and in the US, NatTurner lead his unsuccessful slave rebellion.

In 1833, slavery was abolished in the British Empire, and the next year, Charles Babbage invented a precursorto the computer, the analytical engine.

Significant mathematical progress during Galois’ lifetime

There were several mathematicians who worked around the time of Galois. Perhaps the earliest was Augustin-Louis Cauchy. He was very famous, very eager to push publications. His work was in analysis, algebra, numbertheory, and discrete topology. The criteria for Convergence for infinite series and permutation group theory areboth thanks to him. His research was also inclusive of, but not limited to, differential equations, determinants,and probability. However, he unfortunately mislaid manuscripts of Abel and Galois, two of his successors. Healso received credit for group theory, even though it was Galois who first came up with it. In this way, he andGalois had somewhat of a competitive relationship, rather than a collaborative one.

The next, chronologically, was August Ferdinant Mobius, who, despite being a professor of physics and as-tronomy, made profound contributions to mathematics. His most important contributions were in numbertheory, topology, and projective geometry. He also further explored what came to be known as the Mobiusstrip, among other concepts in projective geometry.

Nicolai Ivonovitch Lobachevsky, from Russia, discovered non-Euclidean geometry. He regarded it as more thatjust theory, believing it to have some real application in the real world. Moreover, he was involved in diversetopics in analysis and physics. Around the same time, Michel Floreal Chasles and Jakob Steiner both madecontributions in synthetic geometry.

A German mathematician, Julius Plucker, invented line geometry enumerative geometry, geometry beyondthree dimensions, and generalizations of projective geometry. His work inspired the creation of modern analyticgeometry, and later mathematicians built on his work. Another German, Jacobi worked in algebra and analysisof complex variables. He also worked in number theory, (cubic reciprocity) succeeding Gauss. His earliest,most important achievement was the theory of elliptic functions, which he then applied to number theory.He famously produced a new proof of Fermat's last conjecture. His other discoveries include theta functions,q-series, and more. Finally, he tried to devise a general solution to the quantic equation, but was not successful.In this regard, he failed where Abel and Galois succeeded.

Finally, Niels Henrik Abel was the first to generally prove Newtons Binomial Theorem. He contributed toalgebraic geometry and the theory of equations and pioneered the idea of inversion. He left behind severalsignificant theorems of analysis. He, along with Galois, was a founder of group theory.

Connections between history and the development of mathematics

It is notable that the majority of the most influential mathematicians around the time of Galois were eitherGerman or French. While this could be due to many reasons, it could be theorized that the prolific contributionsmade by these two countries were a result of the rapid political and (to a lesser extent) intellectual developmentsthey went through. The revolutions in France have already been discussed earlier in this paper, but those inGermany are significant as well.

At this time, Germany was seeing the beginnings of a parliamentary system. The document that establishedthe German Confederation, the Federal Act, outlined the need for constitutions for each individual state. Most

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states acquiesced, with only two states holding out until about 1848. With new representative democracies,parliamentary democracy would have grown. In a country with such a rapidly growing political sphere, itis likely that there would be a greater number of citizens looking to the wise men of the country, such asphilosophers, logicians, politicians, and mathematicians. (Remember that Galois was a heavily politically-involved man) This would result in the expansion of all those roles, including mathematician.

Remarks

It seems that the most interesting part of Galois' story is not so much the contributions that he made (althoughthey are significant) but his life story, which was characterized by political and familial turmoil. Sadly, hereceived little recognition for his work during his lifetime. After his death, however, the usefulness of it wasdiscovered. He lived during a time of incredible political revolution. As a young radical liberalist in a highlyconservative nation, it is no surprise that he ran into so much trouble with his government, especially giventheir tendency to prosecute the dissenters. Galois at times, contrary to what one would expect from such amathematical prodigy, seemed to be quite the hotheaded, even impulsive young man.

References

1. http://www.evariste-galois.net/

2. http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Galois.html

3. http://www.britannica.com/biography/Evariste-Galois

4. http://www.britannica.com/event/Bourbon-Restoration

5. http://www.thepeoplehistory.com/1800s.htmlworld

6. http://fabpedigree.com/james/grmatm4.htm

7. http://fabpedigree.com/james/grmatm5.htm

8. https://www.bundestag.de/

9. http://gonitsora.com/evariste-galois/

10. https://lacamerata.wordpress.com/2013/01/13/the-short-brilliant-life-of-evariste-galois/

11. http://www.bonjourlafrance.com/france-history/french-restoration.htm

12. http://www.globalsecurity.org/military/world/europe/fr-reaction.htm

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