In an often-quoted impressive paper Vandermonde  comes very close to introducing the concept of Cyclotomy. Discussing the work of Vandermonde, Jean-Pierre Tignol  writes
∙∙∙The existence of a cyclic permutation that does preserve the relations (among the roots) is very remarkable; and quite mysterious property of cyclotomic equations, which should have to awaken Vandermonde’s curiosity. If he had investigated this property he could have developed the theory of cyclotomy about 30 years before Gauss.
The fact that Vandermonde did his investigations without the benefit of the knowledge of the work of Gauss (as Galois did) is indeed very noteworthy. His observation about the existence of cyclic permutation which preserves the relations among the roots is forerunner to the introduction of the concept of automorphism in later times. From Phillip S. Jones  we learn what Lebesgue thought about the work of Vandermonde.
∙∙∙Vandermonde’s real and unrecognized claim to fame was lodged in his first paper, in which he approached the general problem of the solvability of algebraic equations through a study of functions invariant under permutations of the roots of the equations.
Several overviews of Vandermonde’s algebraic work have already appeared in print. For an excellent account by Olaf Neumann see . For a short biographical sketch of Vandermonde containing Lebesgue’s opinion about his work see Phillip S. Jones cited above.
In the last article of the aforementioned paper, Vandermonde talks about applying the theory developed earlier to solve (that is to find radical expressions for the roots of) the equation , having disposed of the similar problems for , for smaller values of n in the earlier articles of the same paper. In particular, he gives radical expressions for for (actually for some unknown reasons Vandermonde prefers to work with the equation whose roots are for ). Vandermonde does not provide the details of his calculations—he only points out the method of doing so and gives the final result. Tignol  has reproduced the radical expressions for for as they appear in Vandermonde’s paper—still not providing any of the details which led him (or Vandermonde) to the radical expressions for for . These radical expression for for , given by Vandermonde and reproduced from there by Tignol are:
then the values of are given by the following expression (with appropriate choice of fifth roots of unity)
Despite our repeated efforts, we failed to verify the correctness of above expressions. We decided to evaluate the expressions for and ourselves using the reductions procedure indicated in Vandermonde’s paper and obtained the following values:
The contribution of Vandermonde to the solutions of equations predates the work of Lagrange on the same by some twenty years. The work of Vandermonde and the opinion of Lagrange about it have also been mentioned in the article by Robin Rider Hamburg . Specifically, while discussing cyclotomic equation Lagrange  makes the following statement about solving the equation
By taking u negatively, the signs of all even terms are changed, and we have the equation solved by Vandermonde. This author only gives the expression obtained as a result of his general method, without indicating the details of the operations by which he achieved it, and nobody after him has been concerned, as far as I know, with the verification of this result, which may have remained unknown.
which among other things essentially says that by replacing u by one obtains the equation solved by Vandermonde, namely , and he does not know of anyone who has verified the results obtained by Vandermonde. In fact, Lagrange also carries out the same calculations. The values obtained by us are exactly the same as ones found by Lagrange so we have some reason to have confidence in the correctness of our calculations.
Although the values of and given by Vandermonde appear to be different from ours but surprisingly and are the same as our and respectively. This is so because
as can be easily verified.
Also as it has been pointed out by Lagrange that if in the expressions for and given by Vandermonde the quantities and are interchanged then these coincide with values obtained by him (and us as well) since
2. Radical Expression for via Galois Theory
In the explanation that follows we shall substantially adhere to the notations used by Vandermonde and Tignol. However, at some places, we shall use the notions from elementary Galois Theory for the justification for some of the steps in the calculations.
We begin by considering the equation
Dividing the above by we get the equation
whose roots are
Dividing this equation by and making the substitution we get
whose five roots are:
We would have preferred to work with Equation (3). However as indicated earlier, for some reasons Vandermonde replaces y by −x in (3). In order to provide a convenient comparison with Vandermonde’s calculations, we shall do likewise. So by replacing y by −x in (3) we get the equation
whose five roots are: , for .
If we let , the mapping , which permutes the 10 roots of Equation (2) as
gives a generator of the group of automorphism of the field . The restriction of this mapping to the subfield (its maximum real subfield) induces an automorphism of . To see the effect of this automorphism on we write:
Consider the tower of fields:
The automorphism of the field given by induces an automorphism of its subfield which cyclically permutes the five roots of (3) as follows:
which is the same as
In other words, the cyclic permutation preserves the (rational) relations among the roots of Equation (3). In the words of Tignol
Vandermonde’s brilliant (but not quite explicit) observation is that the permutation preserves the relations between the roots.
As , we have (which is same as ). Applying the permutation to the relation
(which is the same as ).
which is the same as saying
which is one of the 15 relations (among and e) that Vandermonde indicates for the process of reduction (to be explained soon). The following is an easily verifiable list of relations among and e given by Vandermonde
Since the permutation induces the permutations
all the 15 relations among and e given by Vandermonde can be derived by applying the permutation to each of the following three relations,
It can be easily seen that any polynomial in and e with rational coefficients can be reduced to a linear polynomial in the same by using the 15 relations among them given by Vandermonde. We refer to this procedure as the process of reduction.
Let be a primitive fifth root of unity. It is easy to see that
For later use, we shall write down explicitly the values of for .
Since and e are roots of (4) we also have
Let and be defined as follows:
We note that , for lies in the field which is of degree 20 over . It is easily seen that
Since we already have radical expression for , it is enough to give a radical expression for ’s for . This can be done by showing that the fifth powers of ’s lie in the field and we shall have the radical expressions for the roots of (4) as soon as we have an explicit expression for the fifth powers of ’s as members of .
Consider the following diagram of the subfields of
Evidently, the Galois group is cyclic of order 20. Let denote the automorphism of the field which fixes but cyclically permutes the five roots of (4) as follows:
Let denote the automorphism of the field which maps to (and so ) and fixes a (and therefore ). Since and commute and have orders 5 and 4 respectively, the automorphism has order 20 and we have
Now as , we have
Therefore and so the fixed subfield of is . In other words . Likewise .
Table 1. Action of the Group on some generating elements of the field .
Table 2. Galois correspondence between the subgroups of and subfields of .
Although we know that but nevertheless, it is quite cumbersome to find an actual expression for it as an element of . It is hard to imagine how so very patiently Vandermonde may have carried out this computation. While doing calculation by hand it would be unwise to write down the terms of and then attempt to simplify these using the 15 relations from Vandermonde’s paper indicated earlier (and the fact that and ). The right approach would be to find the values of , , and in succession and simplifying each (i.e. expressing it as a linear polynomial in and e with coefficients in ) before attempting to find the next higher power. Undoubtedly, Vandermonde followed this approach although he neither so claims and nor does he give the expressions for , , and in his paper. The fact that Vandermonde’s end result is slightly incorrect does not in any way diminish our appreciation for his patiently carrying out enormous calculation by hand.
Since the automorphism of permutes ’s cyclically as , we only need to compute . We shall not deprive the readers of the joy of calculating by hand. Here we shall content ourselves by giving the explicit expressions for , , and . We have checked their correctness with the help of MATHEMATICA. For the purposes of calculations, we shall find it more convenient to write each power of as a polynomial in with coefficients which are linear in and e.
In a like manner (or preferably by simply applying the automorphism ) we obtain similar expressions for and . These expressions are:
By substituting the values of for in radical form we get the following radical expressions for , for .
We observe that . With appropriate choice of fifth roots of and we can find the radical expressions for the roots of the Equation (4). However to check the correctness of our expressions for and we proceed as follows.
Let and be defined as follows:
The correctness of these expressions can be easily checked with the help of a calculator. Finally, for the curious, we have the following explicit radical expression for .
This indeed gives the correct value of with a proper choice of the fifth roots of unity in the above expression.
We would like to express our thanks and appreciation to two anonymous mathematicians for bringing to our attention references   and . We also thank Professor Clifford Johnston of the Department of Mathematics, West Chester University for his help with MATHEMATICA.
 Vandermonde, A.T. (1771) Mémoire sur la résolution des équations, Histoire de l'Acad. Royale des Sciences (avec les mémoires de Math. & Phys. pour la même année, tirés des registres de cette Acad.), 365-416.
 Neumann, O. (2007) Cyclotomy: From Euler through Vandermonde to Gauss. In: Bradley, R.E. and Sandifer, C.E., Eds., Leonhard Euler: Life, Work and Legacy, Elsevier, Amsterdam, 323-361.