Évariste Galois (/ɡælˈwɑː/; French: [evaʁist ɡalwa]; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem standing for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. He died at age 20 from wounds suffered in a duel.
A portrait of Évariste Galois aged about 15
|Born||25 October 1811|
|Died||31 May 1832 (aged 20)|
Paris, Kingdom of France
|Alma mater||École préparatoire (no degree)|
|Known for||Work on the theory of equations and Abelian integrals|
Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (née Demante). His father was a Republican and was head of Bourg-la-Reine's liberal party. His father became mayor of the village after Louis XVIII returned to the throne in 1814. His mother, the daughter of a jurist, was a fluent reader of Latin and classical literature and was responsible for her son's education for his first twelve years.
He found a copy of Adrien-Marie Legendre's Éléments de Géométrie, which, it is said, he read "like a novel" and mastered at the first reading. At 15, he was reading the original papers of Joseph-Louis Lagrange, such as the Réflexions sur la résolution algébrique des équations which likely motivated his later work on equation theory, and Leçons sur le calcul des fonctions, work intended for professional mathematicians, yet his classwork remained uninspired, and his teachers accused him of affecting ambition and originality in a negative way.
In 1828, he attempted the entrance examination for the École Polytechnique, the most prestigious institution for mathematics in France at the time, without the usual preparation in mathematics, and failed for lack of explanations on the oral examination. In that same year, he entered the École Normale (then known as l'École préparatoire), a far inferior institution for mathematical studies at that time, where he found some professors sympathetic to him.
In the following year Galois' first paper, on continued fractions, was published. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers on this topic to the Academy of Sciences. Augustin-Louis Cauchy refereed these papers, but refused to accept them for publication for reasons that still remain unclear. However, in spite of many claims to the contrary, it is widely held that Cauchy recognized the importance of Galois' work, and that he merely suggested combining the two papers into one in order to enter it in the competition for the Academy's Grand Prize in Mathematics. Cauchy, an eminent mathematician of the time, though with political views that were at the opposite end from Galois', considered Galois' work to be a likely winner.
On 28 July 1829, Galois' father committed suicide after a bitter political dispute with the village priest. A couple of days later, Galois made his second and last attempt to enter the Polytechnique, and failed yet again. It is undisputed that Galois was more than qualified; however, accounts differ on why he failed. More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, which enraged Galois. The recent death of his father may have also influenced his behavior.
Having been denied admission to the Polytechnique, Galois took the Baccalaureate examinations in order to enter the École Normale. He passed, receiving his degree on 29 December 1829. His examiner in mathematics reported, "This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research."
He submitted his memoir on equation theory several times, but it was never published in his lifetime due to various events. Though his first attempt was refused by Cauchy, in February 1830 following Cauchy's suggestion he submitted it to the Academy's secretary Joseph Fourier, to be considered for the Grand Prix of the Academy. Unfortunately, Fourier died soon after, and the memoir was lost. The prize would be awarded that year to Niels Henrik Abel posthumously and also to Carl Gustav Jacob Jacobi. Despite the lost memoir, Galois published three papers that year, one of which laid the foundations for Galois theory. The second one was about the numerical resolution of equations (root finding in modern terminology). The third was an important one in number theory, in which the concept of a finite field was first articulated.
Galois lived during a time of political turmoil in France. Charles X had succeeded Louis XVIII in 1824, but in 1827 his party suffered a major electoral setback and by 1830 the opposition liberal party became the majority. Charles, faced with abdication, staged a coup d'état, and issued his notorious July Ordinances, touching off the July Revolution which ended with Louis-Philippe becoming king. While their counterparts at the Polytechnique were making history in the streets during les Trois Glorieuses, Galois and all the other students at the École Normale were locked in by the school's director. Galois was incensed and wrote a blistering letter criticizing the director, which he submitted to the Gazette des Écoles, signing the letter with his full name. Although the Gazette's editor omitted the signature for publication, Galois was expelled.
Although his expulsion would have formally taken effect on 4 January 1831, Galois quit school immediately and joined the staunchly Republican artillery unit of the National Guard. He divided his time between his mathematical work and his political affiliations. Due to controversy surrounding the unit, soon after Galois became a member, on 31 December 1830, the artillery of the National Guard was disbanded out of fear that they might destabilize the government. At around the same time, nineteen officers of Galois' former unit were arrested and charged with conspiracy to overthrow the government.
In April 1831, the officers were acquitted of all charges, and on 9 May 1831, a banquet was held in their honor, with many illustrious people present, such as Alexandre Dumas. The proceedings grew riotous, and Galois proposed a toast to King Louis Philippe with a dagger above his cup, which was interpreted as a threat against the king's life. He was arrested the following day but was acquitted on 15 June 1831.
On the following Bastille Day (14 July 1831), Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a rifle, and a dagger. He was again arrested. On 23 October, he was sentenced to six months in prison for illegally wearing a uniform. He was released on 29 April 1832. During his imprisonment, he continued developing his mathematical ideas.
Galois returned to mathematics after his expulsion from the École Normale, although he continued to spend time in political activities. After his expulsion became official in January 1831, he attempted to start a private class in advanced algebra which attracted some interest, but this waned, as it seemed that his political activism had priority. Siméon Poisson asked him to submit his work on the theory of equations, which he did on 17 January 1831. Around 4 July 1831, Poisson declared Galois' work "incomprehensible", declaring that "[Galois'] argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor"; however, the rejection report ends on an encouraging note: "We would then suggest that the author should publish the whole of his work in order to form a definitive opinion." While Poisson's report was made before Galois' July 14 arrest, it took until October to reach Galois in prison. It is unsurprising, in the light of his character and situation at the time, that Galois reacted violently to the rejection letter, and decided to abandon publishing his papers through the Academy and instead publish them privately through his friend Auguste Chevalier. Apparently, however, Galois did not ignore Poisson's advice, as he began collecting all his mathematical manuscripts while still in prison, and continued polishing his ideas until his release on 29 April 1832, after which he was somehow talked into a duel.
Galois' fatal duel took place on 30 May. The true motives behind the duel are obscure. There has been much speculation as to the reasons behind it. What is known is that five days before his death, he wrote a letter to Chevalier which clearly alludes to a broken love affair.
Some archival investigation on the original letters suggests that the woman of romantic interest was Stéphanie-Félicie Poterin du Motel, the daughter of the physician at the hostel where Galois stayed during the last months of his life. Fragments of letters from her, copied by Galois himself (with many portions, such as her name, either obliterated or deliberately omitted), are available. The letters hint that du Motel had confided some of her troubles to Galois, and this might have prompted him to provoke the duel himself on her behalf. This conjecture is also supported by other letters Galois later wrote to his friends the night before he died. Galois' cousin, Gabriel Demante, when asked if he knew the cause of the duel, mentioned that Galois "found himself in the presence of a supposed uncle and a supposed fiancé, each of whom provoked the duel." Galois himself exclaimed: "I am the victim of an infamous coquette and her two dupes."
Much more detailed speculation based on these scant historical details has been interpolated by many of Galois' biographers (most notably by Eric Temple Bell in Men of Mathematics), such as the frequently repeated speculation that the entire incident was stage-managed by the police and royalist factions to eliminate a political enemy.
As to his opponent in the duel, Alexandre Dumas names Pescheux d'Herbinville, who was actually one of the nineteen artillery officers whose acquittal was celebrated at the banquet that occasioned Galois' first arrest. However, Dumas is alone in this assertion, and if he were correct it is unclear why d'Herbinville would have been involved. It has been speculated that he might have been du Motel's "supposed fiancé" at the time (she ultimately married someone else), but no clear evidence has been found supporting this conjecture. On the other hand, extant newspaper clippings from only a few days after the duel give a description of his opponent (identified by the initials "L.D.") that appear to more accurately apply to one of Galois' Republican friends, most probably Ernest Duchatelet, who was imprisoned with Galois on the same charges. Given the conflicting information available, the true identity of his killer may well be lost to history.
Whatever the reasons behind the duel, Galois was so convinced of his impending death that he stayed up all night writing letters to his Republican friends and composing what would become his mathematical testament, the famous letter to Auguste Chevalier outlining his ideas, and three attached manuscripts. Mathematician Hermann Weyl said of this testament, "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind." However, the legend of Galois pouring his mathematical thoughts onto paper the night before he died seems to have been exaggerated. In these final papers, he outlined the rough edges of some work he had been doing in analysis and annotated a copy of the manuscript submitted to the Academy and other papers.
Early in the morning of 30 May 1832, he was shot in the abdomen, abandoned by his opponents and seconds, and was found by a passing farmer. He died the following morning at ten o'clock in the Hôpital Cochin (probably of peritonitis), after refusing the offices of a priest. His funeral ended in riots. There were plans to initiate an uprising during his funeral, but during the same time frame the leaders heard of General Jean Maximilien Lamarque's death, and the rising was postponed without any uprising occurring until 5 June. Only Galois' younger brother was notified of the events prior to Galois' death. He was 20 years old. His last words to his younger brother Alfred were:
"Ne pleure pas, Alfred ! J'ai besoin de tout mon courage pour mourir à vingt ans !"
(Don't cry, Alfred! I need all my courage to die at twenty!)
On 2 June, Évariste Galois was buried in a common grave of the Montparnasse Cemetery whose exact location is unknown. In the cemetery of his native town – Bourg-la-Reine – a cenotaph in his honour was erected beside the graves of his relatives.
In 1843 Liouville reviewed his manuscript and declared it sound. It was finally published in the October–November 1846 issue of the Journal de Mathématiques Pures et Appliquées. The most famous contribution of this manuscript was a novel proof that there is no quintic formula – that is, that fifth and higher degree equations are not generally solvable by radicals. Although Abel had already proved the impossibility of a "quintic formula" by radicals in 1824 and Ruffini had published a solution in 1799 that turned out to be flawed, Galois' methods led to deeper research in what is now called Galois theory. For example, one can use it to determine, for any polynomial equation, whether it has a solution by radicals.
From the closing lines of a letter from Galois to his friend Auguste Chevalier, dated May 29, 1832, two days before Galois' death:
Tu prieras publiquement Jacobi ou Gauss de donner leur avis, non sur la vérité, mais sur l'importance des théorèmes.
Après cela, il y aura, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.
(Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)
Within the 60 or so pages of Galois' collected works are many important ideas that have had far-reaching consequences for nearly all branches of mathematics. His work has been compared to that of Niels Henrik Abel, another mathematician who died at a very young age, and much of their work had significant overlap.
While many mathematicians before Galois gave consideration to what are now known as groups, it was Galois who was the first to use the word group (in French groupe) in a sense close to the technical sense that is understood today, making him among the founders of the branch of algebra known as group theory. He developed the concept that is today known as a normal subgroup. He called the decomposition of a group into its left and right cosets a proper decomposition if the left and right cosets coincide, which is what today is known as a normal subgroup. He also introduced the concept of a finite field (also known as a Galois field in his honor), in essentially the same form as it is understood today.
In his last letter to Chevalier and attached manuscripts, the second of three, he made basic studies of linear groups over finite fields:
Galois' most significant contribution to mathematics is his development of Galois theory. He realized that the algebraic solution to a polynomial equation is related to the structure of a group of permutations associated with the roots of the polynomial, the Galois group of the polynomial. He found that an equation could be solved in radicals if one can find a series of subgroups of its Galois group, each one normal in its successor with abelian quotient, or its Galois group is solvable. This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the theory of equations to which Galois originally applied it.
As written in his last letter, Galois passed from the study of elliptic functions to consideration of the integrals of the most general algebraic differentials, today called Abelian integrals. He classified these integrals into three categories.
In his first paper in 1828, Galois proved that the regular continued fraction which represents a quadratic surd ζ is purely periodic if and only if ζ is a reduced surd, that is, and its conjugate satisfies .
In fact, Galois showed more than this. He also proved that if ζ is a reduced quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other. In symbols we have
where ζ is any reduced quadratic surd, and η is its conjugate.
From these two theorems of Galois a result already known to Lagrange can be deduced. If r > 1 is a rational number that is not a perfect square, then
In particular, if n is any non-square positive integer, the regular continued fraction expansion of √n contains a repeating block of length m, in which the first m − 1 partial denominators form a palindromic string.
An algebraic solution or solution in radicals is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots (square roots, cube roots, and other integer roots).
The most well-known example is the solution
introduced in secondary school, of the quadratic equation
(where a ≠ 0).
There exist more complicated algebraic solutions for the general cubic equation and quartic equation. The Abel–Ruffini theorem states that the general quintic equation lacks an algebraic solution, and this directly implies that the general polynomial equation of degree n, for n ≥ 5, cannot be solved algebraically. However, for n ≥ 5, some polynomial equations have algebraic solutions; for example, the equation can be solved as See Quintic function § Other solvable quintics for various other examples in degree 5.
Évariste Galois introduced a criterion allowing one to decide which equations are solvable in radicals. See Radical extension for the precise formulation of his result.
Algebraic solutions form a subset of closed-form expressions, because the latter permit transcendental functions (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.Auguste Bravais
Auguste Bravais (French pronunciation: [oɡyst bʁavɛ]; 23 August 1811, Annonay, Ardèche – 30 March 1863, Le Chesnay, France) was a French physicist known for his work in crystallography, the conception of Bravais lattices, and the formulation of Bravais law. Bravais also studied magnetism, the northern lights, meteorology, geobotany, phyllotaxis, astronomy, and hydrography.
He studied at the Collège Stanislas in Paris before joining the École Polytechnique in 1829, where he was a classmate of groundbreaking mathematician Évariste Galois, whom Bravais actually beat in a scholastic mathematics competition. Towards the end of his studies he became a naval officer, and sailed on the Finistere in 1832 as well as the Loiret afterwards. He took part in hydrographic work along the Algerian Coast. He participated in the Recherche expedition and helped the Lilloise in Spitzbergen and Lapland.
Bravais taught a course in applied mathematics for astronomy in the Faculty of Sciences in Lyon, starting in 1840. He succeeded Victor Le Chevalier in the Chair of Physics at the Ecole Polytechnique between 1845 and 1856, after which he was replaced by Henri Hureau de Sénarmont. He is best remembered for his work on Bravais lattices, particularly his 1848 discovery that there are 14 unique lattices in three-dimensional crystalline systems, correcting the previous scheme, with 15 lattices, conceived by Frankenheim three years before.
Bravais published a memoir about crystallography in 1847. A co-founder of the Société météorologique de France, he joined the French Academy of Sciences in 1854. Bravais also worked on the theory of observational errors, a field in which he is especially known for his 1846 paper "Mathematical analysis on the probability of errors of a point".
The mountain Bravaisberget, in Svalbard, is named after Bravais.Fermat's Last Theorem (book)
Fermat's Last Theorem is a popular science book (1997) by Simon Singh. It tells the story of the search for a proof of Fermat's last theorem, first conjectured by Pierre de Fermat in 1637, and explores how many mathematicians such as Évariste Galois had tried and failed to provide a proof for the theorem. Despite the efforts of many mathematicians, the proof would remain incomplete until as late as 1995, with the publication of Andrew Wiles' proof of the Theorem. The book is the first mathematics book to become a Number One seller in the United Kingdom, whilst Singh's documentary The Proof, on which the book was based, won a BAFTA in 1997.In the United States, the book was released as Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. The book was released in the United States in October 1998 to coincide with the US release of Singh's documentary The Proof about Wiles's proof of Fermat's Last Theorem.Galois geometry
Galois geometry (so named after the 19th century French Mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field). More narrowly, a Galois geometry may be defined as a projective space over a finite field.Objects of study include affine and projective spaces over finite fields and various structures that are contained in them. In particular, arcs, ovals, hyperovals, unitals, blocking sets, ovoids, caps, spreads and all finite analogues of structures found in non-finite geometries. Vector spaces defined over finite fields play a significant role, especially in construction methods.Galois group
In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.
For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.Galois theory
In mathematics, Galois theory provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood.
The subject is named after Évariste Galois, who introduced it for studying the roots
of a polynomial and characterizing the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is solvable by radicals if its roots may be expressed by a formula involving only integers, nth roots and the four basic arithmetic operations.
The theory has been popularized (among mathematicians) and developed by Richard Dedekind, Leopold Kronecker and Emil Artin, and others, who, in particular, interpreted the permutation group of the roots as the automorphism group of a field extension.
Galois theory has been generalized to Galois connections and Grothendieck's Galois theory.List of people killed in duels
This is a list of people killed in duels by date:
Cadeguala, Mapuche toqui, by Alonso García de Ramón at Purén, Chile – 1585
Sir William Drury, English politician and soldier, by Sir John Borough, died from wound received in duel in France – 1590
Gabriel Spenser, Elizabethan actor, by Ben Jonson on Hoxton Fields, London – 1598
Sir John Townsend, English politician, by Sir Thomas Browne on Hounslow Heath, London – 1603
Peter Legh, English politician, by Valentine Browne – 1640
Armand d'Athos, inspiration for the Alexandre Dumas character of the same name – 1643
Charles Price, English politician, by Capt. Robert Sandys at Presteigne – 1645
Sir Henry Bellasis (heir of John Belasyse, 1st Baron Belasyse), by Thomas Porter (dramatist) at Covent Garden, London – 1667
Francis Talbot, 11th Earl of Shrewsbury, by the Duke of Buckingham – 1668
Charles Mohun, 3rd Baron Mohun of Okehampton, acting as second to William Cavendish, 4th Duke of Devonshire – 1677
Walter Norborne, English politician, by an Irishman at the fountain at Middle Temple, London – 1684
John Talbot, brother of the Earl of Shrewsbury, by Henry Fitzroy, 1st Duke of Grafton – 1686
Sir Henry Hobart, English politician, by Oliver Le Neve on Cawston Heath, Norfolk – 1698
Sir John Hanmer, 3rd Baronet, English politician – 1701
Charles Mohun, 4th Baron Mohun, perennial duellist and James Hamilton, 4th Duke of Hamilton, in Hyde Park, London. The Hamilton–Mohun Duel – 1712
Peder Tordenskjold, Norwegian naval officer, by Jakob Axel Staël von Holstein – 1720
George Lockhart, Scottish politician and writer, Jacobite spy – 1731
Richard Nugent, Lord Delvin, by Capt. George Reilly at Marlborough Bowling Green, Dublin – 1761
Button Gwinnett, signer of the Declaration of Independence by Lachlan McIntosh near Savannah, Georgia – 1777
Sir Barry Denny, 2nd Baronet – 1794
Philip Hamilton, son of former U.S. Secretary of the Treasury, Alexander Hamilton, by George I. Eacker, in Weehawken, New Jersey – 1801
Richard Dobbs Spaight, delegate to the Continental Congress and Governor of North Carolina, by John Stanly – 1802
Peter Lawrence Van Allen, lawyer, by William Harris Crawford, future U.S. Secretary of the Treasury, at Fort Charlotte in South Carolina – 1802
Alexander Hamilton, former U.S. Secretary of the Treasury, by U.S. Vice President Aaron Burr, in Weehawken, New Jersey – 1804
Thomas Pitt, 2nd Baron Camelford, English peer and naval officer, by his friend Thomas Best near Holland House, London – 1804
Charles Dickinson, by future U.S. President Andrew Jackson – 1806
Charles Lucas, legislator in Missouri Territory, by U.S. Senator Thomas Hart Benton – 1817
Armistead Thompson Mason, U.S. Senator from Virginia – 1819
Stephen Decatur, American naval hero, by James Barron – 1820
John Scott, founder and editor of the London Magazine – 1821
Joshua Barton, first Missouri Secretary of State – 1823
Henry Wharton Conway, Arkansas politician – 1827
Évariste Galois, mathematician – 1832
Robert Lyon, last Canadian duelling fatality – 1833
Aleksandr Pushkin, Russian poet and writer of the Romantic era, by Georges d'Anthès – 1837
Jonathan Cilley, U.S. Representative from Maine, by William J. Graves – 1838
Mikhail Lermontov, Russian poet and writer of the Romantic era – 1841
George A. Waggaman, U.S. Senator from Louisiana – 1843
John Hampden Pleasants, American newspaper editor; 1846
Edward Gilbert, U.S. newspaper editor, by James W. Denver near Sacramento – 1852
David C. Broderick, U.S. Senator from California – 1859
Lucius M. Walker, Confederate Civil War general – 1863
Ferdinand Lassalle, German socialist leader – 1864
Manuel Corchado y Juarbe, Puerto Rican poet, journalist and politician – 1884
Felice Cavallotti, Italian radical leader – 1898
Euclides da Cunha, Brazilian writer – 1909List of things named after Évariste Galois
These are things named after Évariste Galois (1811–1832), a French mathematician.
Absolute Galois group
Differential Galois theory
Inverse Galois sequence
Inverse Galois problemLycée Flora Tristan (Noisy-le-Grand)
Lycée Flora Tristan is a French senior high school/sixth-form college in Noisy-le-Grand, in the Paris metropolitan area.Lycée Évariste-Galois
Lycée Évariste-Galois may refer to:
Lycée Evariste Galois (Beaumont-sur-Oise) in Beaumont-sur-Oise
Lycée Évariste Galois (Noisy-le-Grand)
Lycée Évariste Galois (Sartrouville)Lycée Évariste Galois (Noisy-le-Grand)
Lycée Évariste Galois is a French senior high school/sixth-form college in Noisy-le-Grand, in the Paris metropolitan area.Lycée Évariste Galois (Sartrouville)
Lycée Évariste Galois is a senior high school/sixth-form college in Sartrouville, Yvelines, France, in the Paris metropolitan area.Nomos Alpha
Nomos Alpha (Greek: Νόμος α΄) is a piece for solo cello composed by Iannis Xenakis in 1965, commissioned by Radio Bremen for cellist Siegfried Palm, and dedicated to mathematicians Aristoxenus of Tarentum, Évariste Galois, and Felix Klein. This piece is an example of a style of music called, by Xenakis, symbolic music – a style of music which makes use of set theory, abstract algebra, and mathematical logic in order to create and analyze musical compositions. Along with symbolic music, Xenakis is known for his development of stochastic music.
During his lifetime, Xenakis was a vocal critic of modern western music, since the development of polyphony for its diminished set of outside-time structures, especially when compared to folk and the Byzantine musical traditions. This perceived incompleteness of western music was the main impetus for the development of symbolic music and for composing Nomos Alpha, his most well known example.
Nomos Alpha consists of 24 sections divided into two layers. The first layer consists of every section not divisible by four, while the second layer consists of every fourth section.Normal subgroup
In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup H of a group G is normal in G if and only if gH = Hg for all g in G. The definition of normal subgroup implies that the sets of left and right cosets coincide. In fact, a seemingly weaker condition that the sets of left and right cosets coincide also implies that the subgroup H of a group G is normal in G. Normal subgroups (and only normal subgroups) can be used to construct quotient groups from a given group.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.Paul Dupuy
Paul du Puy, History Lecturer at the Ecole Normale, published in 1896 the first scientific biography of the mathematician Évariste Galois, titled "La vie d'Évariste Galois".
He attended the École normale supérieure at rue d'Ulm, Paris.
His schoolmates included the future geographers Marcel Dubois and Bertrand Auerbach, and the future historians Georges Lacour-Gayet, Salomon Reinach and Gustave Lanson. He was one of the first teachers to teach at the International School of Geneva, the world's first international school.Primitive permutation group
In mathematics, a permutation group G acting on a non-empty set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X, where nontrivial partition means a partition that isn't a partition into singleton sets or partition into one set X. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive.
While primitive permutation groups are transitive by definition, not all transitive permutation groups are primitive. The requirement that a primitive group be transitive is necessary only when X is a 2-element set and the action is trivial; otherwise, the condition that G preserves no nontrivial partition implies that G is transitive. This is because for non-transitive actions either the orbits of G form a nontrivial partition preserved by G, or the group action is trivial, in which case any nontrivial partition of X (which exists for |X|≥3) is preserved by G.
This terminology was introduced by Évariste Galois in his last letter, in which he used the French term équation primitive for an equation whose Galois group is primitive.In the same letter he stated also the following theorem.
If G is a primitive solvable group acting on a finite set X, then the order of X is a power of a prime number p, X may be identified with an affine space over the finite field with p elements and G acts on X as a subgroup of the affine group.
An imprimitive permutation group is an example of an induced representation; examples include coset representations G/H in cases where H is not a maximal subgroup. When H is maximal, the coset representation is primitive.
If the set X is finite, its cardinality is called the "degree" of G.
The numbers of primitive groups of small degree were stated by Robert Carmichael in 1937:
Note the large number of primitive groups of degree 16. As Carmichael notes, all of these groups, except for the symmetric and alternating group, are subgroups of the affine group on the 4-dimensional space over the 2-element finite field.Resolvent (Galois theory)
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are
These three resolvents have the property of being always separable, which means that, if they have a multiple root, then the polynomial p is not irreducible. It is not known if there is an always separable resolvent for every group of permutations.
For every equation the roots may be expressed in terms of radicals and of a root of a resolvent for a resoluble group, because, the Galois group of the equation over the field generated by this root is resoluble.Sainte-Pélagie Prison
Sainte-Pélagie was a prison in Paris, in active use from 1790 to 1899. The former Parisian prison was located between the current group of buildings bearing No. 56 Rue de la Clef with Rue du Puits-de-l'Ermite in the 5th arrondissement of Paris at the old Place Sainte-Pélagie.
The penal structure held many noted prisoners during the French Revolution, with Madame Roland and Grace Dalrymple Elliott being the only female prisoners. After the revolution, the Marquis de Sade was imprisoned here, as was the young mathematician Évariste Galois. During the July Monarchy, the "April insurgees" were also detained there, and some managed to escape through an underground tunnel. The painter Gustave Courbet was also imprisoned here for his activities in the Paris Commune. He painted a self-portrait titled, Gustave Courbet: Self-Portrait at Sainte-Pélagie.Theory of equations
In algebra, the theory of equations is the study of algebraic equations (also called “polynomial equations”), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an algebraic solution. This problem was completely solved in 1830 by Évariste Galois, by introducing what is now called Galois theory.
Before Galois, there was no clear distinction between the “theory of equations” and “algebra”. Since then algebra has been dramatically enlarged to include many new subareas, and the theory of algebraic equations receives much less attention. Thus, the term "theory of equations" is mainly used in the context of the history of mathematics, to avoid confusion between old and new meanings of “algebra”.