Lagrange resolvent
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 resoluble group, because, the Galois group of the equation over the field generated by this root is resoluble.
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. This defines the generic polynomial of degree n
where Ei is the ithelementary symmetric polynomial.
The symmetric group Sn acts on the Xi by permuting them, and this induces an action on the polynomials in the Xi. The stabilizer of a given polynomial under this action is generally trivial, but some polynomials have a bigger stabilizer. For example, the stabilizer of an elementary symmetric polynomial is the whole group Sn. If the stabilizer is non-trivial, the polynomial is fixed by some non-trivial subgroup G; it is said an invariant of G. Conversely, given a subgroup G of Sn, an invariant of G is a resolvent invariant for G if it is not an invariant of any bigger subgroup of Sn.
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