In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root.
Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois.
Nowadays they are still a fundamental tool to compute Galois groups.
The simplest examples of resolvents are These three resolvents have the property of being always separable, which means that, if they have a multiple root, then the polynomial p is not irreducible.
It is not known if there is an always separable resolvent for every group of permutations.
For every equation the roots may be expressed in terms of radicals and of a root of a resolvent for a solvable group, because the Galois group of the equation over the field generated by this root is solvable.
Let n be a positive integer, which will be the degree of the equation that we will consider, and (X1, ..., Xn) an ordered list of indeterminates.
According to Vieta's formulas this defines the generic monic polynomial of degree n
where Ei is the i th elementary symmetric polynomial.
The symmetric group Sn acts on the Xi by permuting them, and this induces an action on the polynomials in the Xi.
For example, the stabilizer of an elementary symmetric polynomial is the whole group Sn.
[1] Finding invariants for a given subgroup G of Sn is relatively easy; one can sum the orbit of a monomial under the action of Sn.
However, it may occur that the resulting polynomial is an invariant for a larger group.
For example, consider the case of the subgroup G of S4 of order 4, consisting of (12)(34), (13)(24), (14)(23) and the identity (for the notation, see Permutation group).
If P is a resolvent invariant for a group G of index m inside Sn, then its orbit under Sn has order m. Let P1, ..., Pm be the elements of this orbit.
The Galois group of a polynomial of degree
If a polynomial is separable and irreducible, then the corresponding Galois group is a transitive subgroup.