It does not seem to be so well-known that there can be only very few linear relations between the roots of an irreducible polynomial which has a “large” Galois group (in a certain sense, which is made clearer below). For instance, given a monic polynomial P in Q[T], written
it is easy to create a polynomial Q of the form
whose roots generate the same field. If βi are the roots of Q, the vanishing of the coefficient of Td-1 means that they satisfy the Q-linear relation
If the Galois group of the splitting field of Q (or P) is the full symmetric group on d letters, then in fact this is the only possible relation (up to multiplication by a fixed scalar)!
In other words, it is impossible, for instance, that the roots αi of P satisfy
Proving this type of fact can be seen either as a fun exercise in the representation theory of finite groups, or as a good way of motivating its usefulness. (The basic knowledge needed is the existence of decomposition into irreducibles, and unicity of the isotypical space corresponding to a given irreducible representation; the treatment by J-P. Serre is probably the clearest introduction).
Indeed, if the question was presented to a student in the form of showing that either of the previous two relations are incompatible with the αi being roots of a polynomial with symmetric Galois group G=Sd, it is likely that it would be natural to expand the hypothetical relation by stating that, for any σ in G, we have (for the last case, say):
If the Galois group is indeed Sd, we can choose σ so that
and the other roots are fixed; then subtracting the new relation from the first one, we get
which is impossible.
This was particularly easy, but to refute the possibility of all relations (except the sum of the roots being 0), something more systematic is needed. The type of argument used suggests to consider the vector space of all possible relations:
and the basic idea is that G acts on R by its action on the roots, in other words, R is a linear representation of G (over C, or a smaller field if one wants).
If we then take the point of view of trying to identify this representation among the representations of G, we are naturally led to remark that it is defined as a subrepresentation of the permutation representation of G acting on the roots
which, as a representation, doesn’t depend on the particular polynomial (and its roots), but only on the group G, as a transitive permutation group acting on d objects.
So, for any particular transitive group G acting on d objects, one can construct the representation F, decompose it into sums of irreducibles πi (this is a problem of group theory), and then it follows that R can only be isomorphic to a direct sum of those representations occuring in R, with multiplicities at most equal to that in F: if
then R can only be of the form
Then one can look at each possibility in turn, and try to see which ones can actually occur as relations of roots of a polynomial. This is clearly easiest if F has few summands, particularly if there is no multiplicity (i.e., if each ni is 1), for then the subspace corresponding to a summand that occurs in R must be equal to that in F, which is itself uniquely determined.
[If there is multiplicity, say n1=2, then in the π1-isotypic component F1 in F, there are infinitely many subspaces isomorphic to π1, with multiplicity one, for instance
for each (fixed) scalar λ, where V is a fixed subspace of F1, G stable and isomorphic to π1 under this action. Each of these subspaces can lead to different relations.]
In the case of the symmetric group, the decomposition of the natural permutation representation is well-known:
where 1 is the subspace generated by
while the complement subspace
is irreducible. Then, 1 is in R precisely when the sum of the roots is zero (and we already know that this possibility can occur), and π can not be (contained in) a relation module, because in particular
which leads (as before, but now in full generality) to the absurd relation
In my paper on relations between zeros of zeta functions over finite fields, I applied this method to understand the possible additive relations (and the multiplicative ones, using the same technique with the module of multiplicative relations instead of R) when the Galois group is the Weyl group of Sp(2g). Here also the situation was simple, with three summands in F, each with multiplicity 1.
In fact, experimenting with Magma, I can see that most transitive permutation groups of small degree seem to have the property of not having multiplicity in the permutation representation. The first counter-example is the symmetric group S3, acting on itself by multiplication (there is no multiplicity in the natural action on three letters); indeed, if G acts on itself by multiplication, the permutation representation is isomorphic to the regular representation, and each irreducible occurs with multiplicity equal to its dimension, so such a transitive action always has multiplicity if the group is non-abelian. But most transitive groups are given with an action on a smaller set, and one finds that among the 173 non-trivial transitive groups of degree at most 11, only 5 exhibit some multiplicity. (Note that it is not clear at all if this impression can be quantified precisely).
This type of problem has been investigated a lot by K. Girstmair, and this paper of his (in Acta Arithmetica, 1999) is probably the most complete introduction to the subject.