The topic of Galois’ paper concerns the possibility of solving polynomial equations, that is, of equations of the form a1xn + a2xn-1 + …. an = 0. Methods for finding the possible solutions of the equation, using only the operations of addition, subtraction, multiplication, division, raising to power and taking a root of the coefficients, had been known for all cases where n is at most 4 (or as the mathematicians would say, where the degree of the equation was at most 4). In mathematical terms, such methods were known as solving the equation by radicals (radical being the symbol for taking the root). It had also been shown that already when the degree of the equation was 5, there are cases where the equation could not be solved by radicals. What Galois added was a criterion by which all equations solvable by radicals could be recognised.
Galois’ method of finding this criterion was preceded by the idea of Lagrange to study the solutions, or to put it in mathematical parlance, the roots of the equation, before they were actually known. Lagrange had especially considered what happens, when in combinations of these roots, using only addition, subtraction, multiplication and division, the places of these roots were changed - permuted, as they say in mathematics. Now, Lagrange had shown that the structure of the equations had something to do with these permutations, although he had not yet managed to find the complete tale.
Galois notes that if a polynomial equation with degree n has n different roots (the maximal number it could have), one could always form a combination of these roots, where every permutation of the roots would change the value of the combination. With a characteristic leap of thought, Galois just states that some combination of the form Aa + Bb + Cc + … would suffice (a, b, c… being roots of the polynomial and A, B, C … being appropriate integers), without proving this statement or even giving any method how to find such combination (the statement is true, but let’s just take it on faith and not go in full detail how to prove it). Furthermore, he points out that Aa + Bb + Cc + … can be assumed to be irreducible, that is, not expressible as a product of other polynomials.
Another point that Galois makes rather quickly is that all of the individual roots can then be expressed as roots of a polynomial in terms of V, which is the numerical result of the combination of the form above, where the root expressed is, say, the first in the combination. In other words, if V = Aa + Bb + Cc + …, then we can find another polynomial F, such that F(V, a) = 0. Even more, Galois points out, just by permuting the roots a, b, c … in a suitable fashion in the combination, one can find for each root (say, b) a numerical result of a suitable permutation of the combination (say, V’), which can then be used to express the root with the same polynomial F (e.g. F(V’,b) = 0). Note that it is quite arbitrary, what is the order of the roots in the first V: it is the permutation of them to make V’ out of V that is of great importance. Indeed, even the exact values V, V’, V’’ … corresponding to the roots a, b, c … are not as important, Galois notes, as the way in which the roots have to be permuted in order to construct the values V, V’, V’’ … . This group of permutations, or group of the equation, as Galois calls it, has the interesting characteristic that whenever some arbitrary function F(a, b, c …) has a value expressible in the by now familiar terms of addition, subtraction, multiplication and division of the roots, then this function will have the exact same value, if the places of the roots are changed with a permutation in this group.
Galois seems to have thought of his group of the equation as a sort of matrix formed of roots, where one line describes one permutation of the roots. Next, Galois makes the interesting suggestion that while the Vs determining the group of the equation are defined by integer coefficients A, B, C …, we could, as he says, adjoin a new quantity, beside integers, that could take the place of the coefficients (this quantity could be, for instance, an irrational root of some integer). With this addition or adjoinment in place, it might become possible to express the polynomial Aa + Bb + Cc + … as a product of further polynomials. If so happens, the group of the equation, expressed in terms of possible values of Aa + Bb + Cc + …, is, as Galois says, partitioned or decomposed into smaller groups. All of these smaller groups happen to be of the same size and also of the same form in the sense that one subgroup can be turned into another with a rulelike permutation.
Now, if just suitable quantities to be adjoined can be found, the procedure can in principle be continued further. In this case, we can move to polynomials of smaller and smaller degrees and eventually hit the rock bottom, when the corresponding group contains nothing but one row. If this can be done, the original equation will be solvable by radicals.
While the aim of the paper is to find the method for recognising polynomial equations solvable by radicals, Galois himself seems to become more and more interested in the supposed mere means for this goal, namely, the group of the equation and the permutations involved. Indeed, this is the road that the development of mathematics and especially algebra was to take: it was not anymore just a method for solving equations, but a more intricate study of such abstract structures like groups of permutations.
Galois notes that if a polynomial equation with degree n has n different roots (the maximal number it could have), one could always form a combination of these roots, where every permutation of the roots would change the value of the combination. With a characteristic leap of thought, Galois just states that some combination of the form Aa + Bb + Cc + … would suffice (a, b, c… being roots of the polynomial and A, B, C … being appropriate integers), without proving this statement or even giving any method how to find such combination (the statement is true, but let’s just take it on faith and not go in full detail how to prove it). Furthermore, he points out that Aa + Bb + Cc + … can be assumed to be irreducible, that is, not expressible as a product of other polynomials.
Another point that Galois makes rather quickly is that all of the individual roots can then be expressed as roots of a polynomial in terms of V, which is the numerical result of the combination of the form above, where the root expressed is, say, the first in the combination. In other words, if V = Aa + Bb + Cc + …, then we can find another polynomial F, such that F(V, a) = 0. Even more, Galois points out, just by permuting the roots a, b, c … in a suitable fashion in the combination, one can find for each root (say, b) a numerical result of a suitable permutation of the combination (say, V’), which can then be used to express the root with the same polynomial F (e.g. F(V’,b) = 0). Note that it is quite arbitrary, what is the order of the roots in the first V: it is the permutation of them to make V’ out of V that is of great importance. Indeed, even the exact values V, V’, V’’ … corresponding to the roots a, b, c … are not as important, Galois notes, as the way in which the roots have to be permuted in order to construct the values V, V’, V’’ … . This group of permutations, or group of the equation, as Galois calls it, has the interesting characteristic that whenever some arbitrary function F(a, b, c …) has a value expressible in the by now familiar terms of addition, subtraction, multiplication and division of the roots, then this function will have the exact same value, if the places of the roots are changed with a permutation in this group.
Galois seems to have thought of his group of the equation as a sort of matrix formed of roots, where one line describes one permutation of the roots. Next, Galois makes the interesting suggestion that while the Vs determining the group of the equation are defined by integer coefficients A, B, C …, we could, as he says, adjoin a new quantity, beside integers, that could take the place of the coefficients (this quantity could be, for instance, an irrational root of some integer). With this addition or adjoinment in place, it might become possible to express the polynomial Aa + Bb + Cc + … as a product of further polynomials. If so happens, the group of the equation, expressed in terms of possible values of Aa + Bb + Cc + …, is, as Galois says, partitioned or decomposed into smaller groups. All of these smaller groups happen to be of the same size and also of the same form in the sense that one subgroup can be turned into another with a rulelike permutation.
Now, if just suitable quantities to be adjoined can be found, the procedure can in principle be continued further. In this case, we can move to polynomials of smaller and smaller degrees and eventually hit the rock bottom, when the corresponding group contains nothing but one row. If this can be done, the original equation will be solvable by radicals.
While the aim of the paper is to find the method for recognising polynomial equations solvable by radicals, Galois himself seems to become more and more interested in the supposed mere means for this goal, namely, the group of the equation and the permutations involved. Indeed, this is the road that the development of mathematics and especially algebra was to take: it was not anymore just a method for solving equations, but a more intricate study of such abstract structures like groups of permutations.
