# Ann Varela: Galois- Mathematician Who Died in a Duel

An Interview with Ann Varela: Galois- Mathematician Who Died in a Duel

Michael F. Shaughnessy –

1) Éveriste Galois was somewhat of a very precocious youth and had some turbulent times during his life. Where was he born, and what were some of his early challenges?

Galois was born in 1811 in Bourg-la-Reine, French Empire. Sources seem to agree that Galois was raised by two wealthy parents who were well-educated. His mother, Adelaide Marie Demante, was his primary tutor until the age of twelve. She was well versed in philosophy, classical literature, and religious studies. His father, Nicholas Gabriel Galois, was a republican and head of Bourg-la-Reine’s liberal party, as well as mayor of the town.

Attending Lycée Louis-le-Grand was phase two of Galois’s education. He maintained enough focus to excel in Latin and won first prize for his efforts. By the age of fourteen, Galois’s interest in most subjects waned, yet his attentiveness to mathematics was intensifying as he read the works of Lagrange, which were thought to inspire his later efforts in equation theory. He also read about geometry in Adrien Marie Legendre’s *Éléments de Géométrie*.

Galois’s inability to stay focused on his work, lack of preparation in broad subject matter, along with poor explanations on the oral examination may have caused him to fail the entrance exam to École Polytechnique. He found some professors compassionate to his situation at the École Normale later that year. Galois attempted another oral entrance exam at École Polytechnique to no avail. Perhaps the death of his father put him in too frail of a state to perform his best.

Another reason rumored to be the cause of his failure is that the examiner simply could not understand the advanced ideas of Galois. To be fair, it is widely noted that Galois’s work tended to lack explanation and instead, jumped from theory to solution with few steps in between all too often.

Contributing to Galois’s frustration was the ostensibly never-ending incompetence or carelessness of his colleagues and other distinguished mathematicians. For example, Cauchy supposedly lost a submitted manuscript of Galois’s that had been presented at the Academy, where yearly prizes for excellence in mathematical pursuits were awarded.

Fourier, who was the secretary of the Academy and appointed to the committee to judge the Grand Prize, received Galois’s manuscript, but died not long afterwards. Consequently, another manuscript was lost. Fourier, however, was not the sole person on the evaluating committee to see this manuscript. Apparently, Lacroix, Poisson, Legendre, and Poinsot were privy to Galois manuscript because they too were members of the prize committee.

In a third and final effort to present his work, Poisson, nonetheless rejected, Galois’s paper. Poisson’s reasoning was based primarily on the manuscript’s lack of supporting proofs, which seems to be a familiar theme with Galois’s work.

2) Polynomial equations—-what are they, and why are they important?

A polynomial is an expression consisting of constants and variables, using only addition, subtraction, or multiplication. Any exponents used must be non-negative whole numbers. A polynomial equation is simply a polynomial expression set equal to zero. Examples of polynomial equations may be seen in Figure 1.

Figure 1. Polynomial Equations.

https://upload.wikimedia.org/wikipedia/commons/a/a3/Latex_math_example_gather_asterisk.png

Polynomial equations play an important role in solving real-life problems, as they can model many situations. They also may be used to form a polynomial function. There are numerous examples of how polynomials are used in real-world situations. Among them, include the topics of chemistry, physics, economics, and numerical analysis. Therefore, the next time you have your vehicle serviced, pay your cable bill, launch a model rocket, or wonder about how to maximize profit for your business, you may be using polynomial equations without even realizing it.

3) Abstract algebra- and group theory seem to be his two main areas of interest. What were his contributions?

Reportedly, Galois was the first mathematician to mention the word *group* in his manuscripts. The circumstances in which he used the term were quite similar to how it is currently used in the branch of mathematics known as group theory. He also pioneered the mathematical principle that is currently known as a normal subgroup, applied in abstract algebra. Galois’s ideas on group theory essentially offer an explanation and specific conditions under which a subgroup is considered to be *normal*. Furthermore, he discusses how an equation must have a chain of normal subgroups to be solvable.

4) Apparently, the lives of Alexandre Dumas (the author of *The Three Musketeers*) and Galois intersect on occasion. What do the historical records tell us about this?

Alexandre Dumas recounted in his memoirs that roughly two hundred republicans gathered at the restaurant *Vendanges des Bourgogne* to celebrate the acquittal of nineteen republicans on conspiracy charges. Galois, with dagger in hand, allegedly uttered something like, “To Louis Phillipe, if he betrayed!” Apparently, the last few words were not completely heard by the crowd due to the noise level of the patrons in attendance. A major misunderstanding may have unfolded and ended with Galois under arrest on the day following the banquet. During his defense, Galois wanted to make it evident to the court that the king was capable of betraying them and further reinforced this belief by citing the current trend in government supported his claim.

5) What are “continued fractions”, and what did Galois have to do with these?

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, etcetera. See Figure 2. Continued fractions give us the “best approximation” of irrational numbers, like the square root of 2. As one can see in Figure 2, continued fractions are made up of fractions. Every number can be written as a continued fraction, whether it is rational or irrational.

Figure 2. Continued Fraction.

A surd is a number or quantity that cannot be expressed as the quotient of two integers. Thus, a number that cannot be simplified to remove a square root (or *n* root) is called a surd. For example, since the square root of 2 only has factors of (square root 2) and (square root 1), the square root of 2 is called a surd. In another example, given the square root of 9, we may break that down into the identical factors of (square root 3) and (square root 3). Since the factors are identical (perfect square factors) and we are dealing with square roots, we say that the square root of 9 is not a surd.

Galois proved in his first paper, written in 1828, that the regular continued fraction, which represents a quadratic surd, is recurrent if and only if the quadratic surd is a reduced surd. See Figure 3.

Figure 3. Quadratic Surd.

6) Theory of equations—why is this important, and what can we learn from these ideas?

Galois’s theory of equations deals with the study of algebraic equations (polynomial equations) which are equations defined by a polynomial. His query dealt with knowing when or if an algebraic equation had an algebraic solution. Thanks to Galois, the query was solved in 1830 using Galois Theory. Galois’s objective was to solve an algebraic equation in terms of radicals, hence, to express the solutions with a formula, which consists of the four operations of arithmetic (addition, subtraction, multiplication, and division) and *n*th roots.

7) There is no quintic formula–Why is this important in regards to Galois?

Galois Theory shows how polynomials of the fifth degree cannot be written down using *n*th roots and the four arithmetic operations. Hence, he proved that there can be no general formula for solving quintic equations. Galois achieved this general proof by looking at whether or not the permutation group of its roots had a certain structure. Permutation groups represent any change in the ordered arrangement of *n* objects. Galois’s theory can provide an algorithm for actually finding the roots of an equation, when these are written in radical notation. Galois’s efforts to solve the quintic polynomial showed that a method other than one involving rational operations and extracting roots would be required to solve a quintic polynomial.

8) Apparently, the night before the duel in which he died- he wrote most of his mathematical concerns. What did he feel was important to include in that letter?

Auguste Chevalier was the recipient of Galois’s famous pre-duel letter. In the letter were mathematical discoveries including new theorems, innovations, and speculations, yet to be shared with other mathematical minds. Therefore, he also requested that the letter be published in the *Revue Encyclopédique* because he felt it was a way for Jacobi and Gauss to express their views of the manuscript’s significance publically.

9) What have I neglected to ask?

Galois seemed to admire his father. Unfortunately, when Galois was 18, his father committed suicide after some political disagreement with a priest. Galois must have felt devastated.

10) Legend or perhaps history has it that he died in a duel- unrequited love perhaps? What do historians seem to gather about his death at the age of 20?

It seems the verdict is still out on the exact circumstances surrounding Galois’s death by bullet in Paris. Possible causes include being shot by a rival in a feud over a woman, being challenged by a royalist who objected to his political views, or that a representative of the police killed him. Apparently, Galois was a member of the National Guard (a pre-republican organization), spent time in prison, and was accused of harassing King Louis Phillippe at one point. So, perhaps some did feel threatened by him.

Name should be with an A not an E Evariste