# The development of Galois theory

### Fiona Brunk

### Évariste Galois

From adolescence onwards, the short life of Évariste Galois (1811-1832) was dominated by dramatic events. What embittered him most was the suicide of his father and several rejections of his mathematical work by the influential French Academy of Science. The fact that Galois spent a considerable amount of the time during which he did mathematics in prison -- as a Republican during the French Revolution -- also influenced at least the presentation of his mathematics. Furthermore, had Galois lived longer, he could have published and clarified more of his theory. However, we will concentrate on his mathematics here; for more information on his biography see Rothman [5] or O'Connor and Robertson [4] and the further reading suggested there.

Galois was the first mathematician whose work clearly demonstrates that he had a concept of a group. Like Lagrange, Galois studied certain arrangements of letters and the permutations on these. Although Galois was not consistent as to whether he regarded the arrangements or the permutations as the group elements, he clearly saw a group as a set with a closure property and worked with normal subgroups. Kiernan [3] points out the possibility that the group concept was actually due to Louis-Paul-Emile Richard (1795-1849) who was Galois' high school teacher at the College Louis-le-Grand which Galois attended between 1823 and 1829. Richard taught many important mathematicians of the time. One of them was Charles Hermite (1822-1882) who introduced Galois Theory to German universities. Hermite demonstrated an understanding of the group concept and familiarity with Galois' papers very soon after their publication. This was so unusual that it leads Kiernan to question whether Galois' ideas were entirely original. However, it seems to me equally possible that Galois discussed his own ideas with Richard who then passed them on to Hermite.

Galois presented many of his ideas with insufficient proofs and in unsuitable terminology. What we call Galois Theory today was largely elaborated and justified by later mathematicians. However, it remains based on two papers by Galois,

Presenting one of Galois' mathematical arguments in full and translating it into today's terminology would be beyond the scope of this summary. As an example, we will merely look at Galois' definition of the group of an equation. In the following quotation [2] of Galois, note that "permutations" are what we call arrangements today, whereas a "substitution" is a permutation and "function of" should be read as "expression in".

In today's terms, when Galois calls an expression "rationally known" he means that it is rational in the elements of a field which has been constructed. In this context, an expression being "invariant" refers to its numerical value; it does not mean "formally unaltered".

Thus we can see that Galois' definition of the group which is so central to his theory uses Lagrange's terminology while at the same time being equivalent to today's definition of the Galois group. This is representative of most of Galois' mathematical work in the sense that it is not immediately recognisable today because it is expressed in terms of 19

The reason Galois' papers were received with interest at the time of publication was that mathematicians hoped to extract from it a procedure for determining whether a given special equation was solvable by radicals and, if so, to solve it. This was, of course, never Galois' intention -- he was not even concerned with practical procedures for determining the group of a given special equation, let alone the solutions of the equation. He did give procedures for determining whether or not a given special equation was solvable by radicals, but they were not practical since they could only be carried out once the roots were known. However, the initial reduction of Galois Theory to its application to the solvability of equations was not entirely due to a misinterpretation of Galois' published work. In

Galois aimed to shift the emphasis of algebra from application to theory: Rather than viewing the theory as necessary to achieve the application, he states that his theory could only be properly understood with an application in mind, and makes many passionate pleas for less direct computation and more interest in structure in mathematical research [3]:

Galois was the first mathematician whose work clearly demonstrates that he had a concept of a group. Like Lagrange, Galois studied certain arrangements of letters and the permutations on these. Although Galois was not consistent as to whether he regarded the arrangements or the permutations as the group elements, he clearly saw a group as a set with a closure property and worked with normal subgroups. Kiernan [3] points out the possibility that the group concept was actually due to Louis-Paul-Emile Richard (1795-1849) who was Galois' high school teacher at the College Louis-le-Grand which Galois attended between 1823 and 1829. Richard taught many important mathematicians of the time. One of them was Charles Hermite (1822-1882) who introduced Galois Theory to German universities. Hermite demonstrated an understanding of the group concept and familiarity with Galois' papers very soon after their publication. This was so unusual that it leads Kiernan to question whether Galois' ideas were entirely original. However, it seems to me equally possible that Galois discussed his own ideas with Richard who then passed them on to Hermite.

Galois presented many of his ideas with insufficient proofs and in unsuitable terminology. What we call Galois Theory today was largely elaborated and justified by later mathematicians. However, it remains based on two papers by Galois,

*Mémoire sur les conditions de résolubilité des équations par radicaux*and*Des équations primitives qui sont solubles par radicaux*, both of which were published in 1846, 14 years after his death.Presenting one of Galois' mathematical arguments in full and translating it into today's terminology would be beyond the scope of this summary. As an example, we will merely look at Galois' definition of the group of an equation. In the following quotation [2] of Galois, note that "permutations" are what we call arrangements today, whereas a "substitution" is a permutation and "function of" should be read as "expression in".

*Let an equation be given whose m roots are a, b, c, ... . There will always be a group of permutations of the letters a, b, c, ... which has the following property:*

*that every function of the roots, invariant under the substitutions of that group, is rationally known;*

*conversely, that every function of the roots, which can be expressed rationally, is invariant under these substitutions.*

In today's terms, when Galois calls an expression "rationally known" he means that it is rational in the elements of a field which has been constructed. In this context, an expression being "invariant" refers to its numerical value; it does not mean "formally unaltered".

Thus we can see that Galois' definition of the group which is so central to his theory uses Lagrange's terminology while at the same time being equivalent to today's definition of the Galois group. This is representative of most of Galois' mathematical work in the sense that it is not immediately recognisable today because it is expressed in terms of 19

^{th}century concepts. However, it is equivalent to the Galois Theory taught today, nearly 200 years after Galois performed his research.The reason Galois' papers were received with interest at the time of publication was that mathematicians hoped to extract from it a procedure for determining whether a given special equation was solvable by radicals and, if so, to solve it. This was, of course, never Galois' intention -- he was not even concerned with practical procedures for determining the group of a given special equation, let alone the solutions of the equation. He did give procedures for determining whether or not a given special equation was solvable by radicals, but they were not practical since they could only be carried out once the roots were known. However, the initial reduction of Galois Theory to its application to the solvability of equations was not entirely due to a misinterpretation of Galois' published work. In

*Des équations primitives...*, his aim was to determine actual criteria which would enable anyone to tell from the coefficients of an equation whether it was solvable by radicals. This aim would of course ultimately fail, but in pursuing it, Galois developed group theory which was much more sophisticated than that previously presented in the*Mémoire*.Galois aimed to shift the emphasis of algebra from application to theory: Rather than viewing the theory as necessary to achieve the application, he states that his theory could only be properly understood with an application in mind, and makes many passionate pleas for less direct computation and more interest in structure in mathematical research [3]:

*Go to the roots of these calculations! Group the operations. Classify them according to their complexities rather than their appearances! This, I believe, is the mission of future mathematicians. This is the road on which I am embarking in this work.*