Évariste Galois

# Évariste Galois

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Évariste Galois (October 25, 1811 – May 31, 1832) was a French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

born in Bourg-la-Reine
Bourg-la-Reine
Bourg-la-Reine is a commune in the southern suburbs of Paris, France. It is located from the center of Paris. The inhabitants are called Réginaburgiens. The town is twinned with Kenilworth, UK.-History:...

. While still in his teens, he was able to determine a necessary and sufficient condition
Necessary and sufficient conditions
In logic, the words necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.-Definitions:A necessary condition...

for a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

, thereby solving a long-standing problem. His work laid the foundations for Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

and group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, two major branches of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, and the subfield of Galois connection
Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...

s. He was the first to use the word "group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

" as a technical term in mathematics to represent a group of permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

The term Radical was used during the late 18th century for proponents of the Radical Movement. It later became a general pejorative term for those favoring or seeking political reforms which include dramatic changes to the social order...

during the monarchy of Louis Philippe
Louis-Philippe of France
Louis Philippe I was King of the French from 1830 to 1848 in what was known as the July Monarchy. His father was a duke who supported the French Revolution but was nevertheless guillotined. Louis Philippe fled France as a young man and spent 21 years in exile, including considerable time in the...

in France, he died from wounds suffered in a duel
Duel
A duel is an arranged engagement in combat between two individuals, with matched weapons in accordance with agreed-upon rules.Duels in this form were chiefly practised in Early Modern Europe, with precedents in the medieval code of chivalry, and continued into the modern period especially among...

under questionable circumstances at the age of twenty.

### Early life

Galois was born on October 25, 1811, to Nicolas-Gabriel Galois and Adélaïde-Marie (born Demante). His father was a Republican and was head of Bourg-la-Reine's liberal party
Liberal Party
Liberal Party is the name for dozens of political parties around the world. Liberal parties can be center-left, centrist, or center-right depending on their location...

. He became mayor of the village after Louis XVIII
Louis XVIII of France
Louis XVIII , known as "the Unavoidable", was King of France and of Navarre from 1814 to 1824, omitting the Hundred Days in 1815...

returned to the throne in 1814. His mother, the daughter of a jurist
Jurist
A jurist or jurisconsult is a professional who studies, develops, applies, or otherwise deals with the law. The term is widely used in American English, but in the United Kingdom and many Commonwealth countries it has only historical and specialist usage...

, was a fluent reader of Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

and classical literature and was responsible for her son's education for his first twelve years. At the age of 10, Galois was offered a place at the college of Reims, but his mother preferred to keep him at home.

In October 1823, he entered the Lycée Louis-le-Grand
Lycée Louis-le-Grand
The Lycée Louis-le-Grand is a public secondary school located in Paris, widely regarded as one of the most rigorous in France. Formerly known as the Collège de Clermont, it was named in king Louis XIV of France's honor after he visited the school and offered his patronage.It offers both a...

, and despite some turmoil in the school at the beginning of the term (when about a hundred students were expelled), Galois managed to perform well for the first two years, obtaining the first prize in Latin. He soon became bored with his studies, and at the age of 14, began to take a serious interest in mathematics.

He found a copy of Adrien Marie Legendre's Éléments de Géométrie, which it is said that he read "like a novel" and mastered at the first reading. At 15, he was reading the original papers of Joseph Louis Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

, such as the landmark 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.

### Budding mathematician

In 1828, he attempted the entrance exam to École Polytechnique
École Polytechnique
The École Polytechnique is a state-run institution of higher education and research in Palaiseau, Essonne, France, near Paris. Polytechnique is renowned for its four year undergraduate/graduate Master's program...

, 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
An Academy of Sciences is a national academy or another learned society dedicated to sciences.In non-English speaking countries, the range of academic fields of the members of a national Academy of Science often includes fields which would not normally be classed as "science" in English...

. Augustin Louis Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...

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 appears 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, a highly eminent mathematician of the time, considered Galois' work to be a likely winner.

On July 28, 1829, Galois' father committed suicide
Suicide
Suicide is the act of intentionally causing one's own death. Suicide is often committed out of despair or attributed to some underlying mental disorder, such as depression, bipolar disorder, schizophrenia, alcoholism, or drug abuse...

after a bitter political dispute with the village priest. A couple of days later, Galois made his second and last attempt at entering the Polytechnique, and failed yet again. It is undisputed that Galois was more than qualified; however, accounts differ on why he failed. The legend holds that he thought the exercise proposed to him by the examiner to be of no interest, and, in exasperation, threw at the examiner's head the rag used to erase the blackboard. More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, evoking the student's rage. 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 December 29, 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. As noted before, his first attempt was refused by Cauchy, but he tried again in February 1830 after following Cauchy's suggestions and submitted it to the Academy's secretary Fourier
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...

, 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 Abel posthumously and also to Jacobi
Carl Gustav Jakob Jacobi
Carl Gustav Jacob Jacobi was a German mathematician, widely considered to be the most inspiring teacher of his time and is considered one of the greatest mathematicians of his generation.-Biography:...

. Despite the lost memoir, Galois published three papers that year, one laid the foundations for Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

,
the second one about the numerical resolution of equations (root finding in modern terminology), and the third, an important one on number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, where the concept of a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

was first articulated.

### Political firebrand

Galois lived during a time of political turmoil in France. Charles X
Charles X of France
Charles X was known for most of his life as the Comte d'Artois before he reigned as King of France and of Navarre from 16 September 1824 until 2 August 1830. A younger brother to Kings Louis XVI and Louis XVIII, he supported the latter in exile and eventually succeeded him...

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
July Ordinances
July Ordinances, also known as the Four Ordinances of Saint-Cloud, were a series of decrees set forth by Charles X and Jules Armand de Polignac, the chief minister, in July 1830....

, touching off the July Revolution
July Revolution
The French Revolution of 1830, also known as the July Revolution or in French, saw the overthrow of King Charles X of France, the French Bourbon monarch, and the ascent of his cousin Louis-Philippe, Duke of Orléans, who himself, after 18 precarious years on the throne, would in turn be overthrown...

which ended with Louis-Philippe
Louis-Philippe of France
Louis Philippe I was King of the French from 1830 to 1848 in what was known as the July Monarchy. His father was a duke who supported the French Revolution but was nevertheless guillotined. Louis Philippe fled France as a young man and spent 21 years in exile, including considerable time in the...

becoming king. While their counterparts at 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 Gazettes editor redacted the signature for publication, Galois was expelled.

Although his expulsion would have formally taken effect on January 4, 1831, Galois quit school immediately and joined the staunchly Republican artillery unit of the National Guard
National Guard (France)
The National Guard was the name given at the time of the French Revolution to the militias formed in each city, in imitation of the National Guard created in Paris. It was a military force separate from the regular army...

. These and other political affiliations continually distracted him from mathematical work. Due to controversy surrounding the unit, soon after Galois became a member, on December 31, 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, the officers were acquitted of all charges, and on May 9, 1831, a banquet was held in their honor, with many illustrious people present, such as Alexandre Dumas
Alexandre Dumas, père
Alexandre Dumas, , born Dumas Davy de la Pailleterie was a French writer, best known for his historical novels of high adventure which have made him one of the most widely read French authors in the world...

. The proceedings grew riotous, and Galois proposed a toast
Toast (honor)
A toast is a ritual in which a drink is taken as an expression of honor or goodwill. The term may be applied to the person or thing so honored, the drink taken, or the verbal expression accompanying the drink. Thus, a person could be "the toast of the evening," for whom someone "proposes 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 June 15.

On the following Bastille Day
Bastille Day
Bastille Day is the name given in English-speaking countries to the French National Day, which is celebrated on 14 July of each year. In France, it is formally called La Fête Nationale and commonly le quatorze juillet...

, 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. For this, he was again arrested and this time was sentenced to six months in prison for illegally wearing a uniform. He was released on April 29, 1832. During his imprisonment, he continued developing his mathematical ideas.

### Final days

Galois returned to mathematics after his expulsion from the École Normale, although constantly distracted by his 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
Activism
Activism consists of intentional efforts to bring about social, political, economic, or environmental change. Activism can take a wide range of forms from writing letters to newspapers or politicians, political campaigning, economic activism such as boycotts or preferentially patronizing...

had priority. Simeon Poisson asked him to submit his work on the theory of equations
Theory of equations
In mathematics, the theory of equations comprises a major part of traditional algebra. Topics include polynomials, algebraic equations, separation of roots including Sturm's theorem, approximation of roots, and the application of matrices and determinants to the solving of equations.From the point...

, which he did on January 17. Around July 4, 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' Bastille Day 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, began collecting all his mathematical manuscripts while still in prison, and continued polishing his ideas until his release on April 29, 1832.

Galois' fatal duel took place on May 30. The true motives behind the duel will most likely remain forever obscure. There has been much speculation, much of it spurious, 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 a Mademoiselle 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 either obliterated, such as her name, or deliberately omitted) are available. The letters hint that Mlle. du Motel had confided some of her troubles with 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. Much more detailed speculation based on these scant historical details has been interpolated by many of Galois' biographers (most notably by Eric Temple Bell
Eric Temple Bell
Eric Temple Bell , was a mathematician and science fiction author born in Scotland who lived in the U.S. for most of his life...

in Men of Mathematics
Men of Mathematics
Men of Mathematics is a book on the history of mathematics written in 1937 by the mathematician E.T. Bell. After a brief chapter on three ancient mathematicians, the remainder of the book is devoted to the lives of about forty mathematicians who worked in the seventeenth, eighteenth and nineteenth...

), 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, one of the nineteen artillery officers whose acquittal was celebrated at the banquet that occasioned Galois' first arrest and du Motel's fiancé. However, Dumas is alone in this assertion, and extant newspaper clippings from only a few days after the duel give a description of his opponent that more accurately applies 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. Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...

, a mathematician, 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 May 30, 1832, he was shot in the abdomen and died the following day at ten in the Cochin hospital (probably of peritonitis
Peritonitis
Peritonitis is an inflammation of the peritoneum, the serous membrane that lines part of the abdominal cavity and viscera. Peritonitis may be localised or generalised, and may result from infection or from a non-infectious process.-Abdominal pain and tenderness:The main manifestations of...

) after refusing the offices of a priest. He was 20 years old. His last words
Last words
Last words are a person's final words spoken before death.Last Words may also refer to:* Last Words , an Australian punk band* Last Words , a memoir by George Carlin* Last Words , a 1968 short film directed by Werner Herzog...

to his 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 June 2, Évariste Galois was buried in a common grave of the Montparnasse cemetery
Montparnasse Cemetery
Montparnasse Cemetery is a cemetery in the Montparnasse quarter of Paris, part of the city's 14th arrondissement.-History:Created from three farms in 1824, the cemetery at Montparnasse was originally known as Le Cimetière du Sud. Cemeteries had been banned from Paris since the closure, owing to...

whose exact location is unknown. In the cemetery of his native town - Bourg-la-Reine
Bourg-la-Reine
Bourg-la-Reine is a commune in the southern suburbs of Paris, France. It is located from the center of Paris. The inhabitants are called Réginaburgiens. The town is twinned with Kenilworth, UK.-History:...

- a cenotaph
Cenotaph
A cenotaph is an "empty tomb" or a monument erected in honour of a person or group of people whose remains are elsewhere. It can also be the initial tomb for a person who has since been interred elsewhere. The word derives from the Greek κενοτάφιον = kenotaphion...

in his honour was erected beside the graves of his relatives.

Galois' mathematical contributions were published in full in 1843 when Liouville
Joseph Liouville
- Life and work :Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris, he was appointed as professor at the École Polytechnique in 1838...

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
Journal de Mathématiques Pures et Appliquées
The Journal de Mathématiques Pures et Appliquées is a French monthly scientific journal of mathematics, founded in 1836 by Joseph Liouville. It is published by Elsevier. According to the 2008 Journal Citation Reports, its impact factor is 1.204. Articles are written in English or French.- External...

. The most famous contribution of this manuscript was a novel proof that there is no quintic formula
Quintic equation
In mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...

, that is, that fifth and higher degree equations are not solvable by radicals. Although Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...

Paolo Ruffini
Paolo Ruffini was an Italian mathematician and philosopher.By 1788 he had earned university degrees in philosophy, medicine/surgery, and mathematics...

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.

## Contributions to mathematics

Unsurprisingly, Galois' collected works amount to only some 60 pages, but within them 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
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...

, another mathematician who died at a very young age, and much of their work had significant overlap.

### Algebra

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
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

. He developed the concept that is today known as a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

. He called the decomposition of a group into its left and right coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...

s 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
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

(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:
• He constructed the general linear group over a prime field, GL(ν, p) and computed its order, in studying the Galois group of the general equation of degree pν.
• He constructed the projective special linear group PSL(2,p). Galois constructed them as fractional linear transforms, and observed that they were simple except if p was 2 or 3. These were the second family of finite simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...

s, after the alternating groups.
• He noted the exceptional fact
Exceptional object
Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects as well as a finite number of exceptions that don't fit into any series. These are known as exceptional...

that PSL(2,p) is simple and acts on p points if and only if p is 5, 7, or 11.

### Galois theory

Galois' most significant contribution to mathematics by far is his development of Galois theory. He realized that the algebraic solution to a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

equation is related to the structure of a group of permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

s associated with the roots of the polynomial, the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

of the polynomial. He found that an equation could be solved in radicals
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

if one can find a series of subgroups of its Galois group, each one normal in its successor with abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

quotient, or its Galois group is solvable
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...

. This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the theory of equations
Theory of equations
In mathematics, the theory of equations comprises a major part of traditional algebra. Topics include polynomials, algebraic equations, separation of roots including Sturm's theorem, approximation of roots, and the application of matrices and determinants to the solving of equations.From the point...

to which Galois originally applied it.

### Analysis

Galois also made some contributions to the theory of Abelian integrals and continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

s.