Resolvent (Galois theory)

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.
Definition.
Let be a positive integer, which will be the degree of the equation that we will consider, and an ordered list of indeterminates. This defines the "generic polynomial" of degree 
where is the "i"th elementary symmetric polynomial.
The symmetric group acts on the by permuting them, and this induces an action on the polynomials in the . 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 . If the stabilizer is non-trivial, the polynomial is fixed by some non-trivial subgroup ; it is said an "invariant" of . Conversely, given a subgroup of , an invariant of is a resolvent invariant for if it is not an invariant of any bigger subgroup of .
Finding invariants for a given subgroup of is relatively easy; one can sum the orbit of a monomial under the action of . However it may occur that the resulting polynomial is an invariant for a larger group. For example, consider the case of the subgroup of of order 4, consisting of , , and the identity (for the notation, see Permutation group). The monomial gives the invariant . It is not a resolvent invariant for , as being invariant by , in fact, it is a resolvent invariant for the dihedral subgroup , and is used to define the resolvent cubic of the quartic equation.
If is a resolvent invariant for a group of index , then its orbit under has order . Let , ..., be the elements of this orbit. Then the polynomial
is invariant under . Thus, when expanded, its coefficients are polynomials in the that are invariant under the action of the symmetry group and thus may be expressed as polynomials in the elementary symmetric polynomials. In other words, is an irreducible polynomial in whose coefficients are polynomial in the coefficients of . Having the resolvent invariant as a root, it is called a resolvent (sometimes resolvent equation).
Consider now an irreducible polynomial
with coefficients in a given field (typically the field of rationals) and roots in an algebraically closed field extension. Substituting the by the and the coefficients of by those of in what precedes, we get a polynomial formula_6, also called "resolvent" or "specialized resolvent" in case of ambiguity). If the Galois group of is contained in , the specialization of the resolvent invariant is invariant by and is thus a root of formula_6 that belongs to (is rational on ). Conversely, if formula_6 has a rational root, which is not a multiple root, the Galois group of is contained in .
Terminology.
There are some variants in the terminology.
Resolvent method.
The Galois group of a polynomial of degree formula_11 is formula_12 or a proper subgroup of that. If a polynomial is separable and irreducible, then the corresponding Galois group is a transitive subgroup.
Transitive subgroups of formula_12 form a directed graph: one group can be a subgroup of several groups. One resolvent can tell if the Galois group of a polynomial is a (not necessarily proper) subgroup of given group. The resolvent method is just a systematic way to check groups one by one until only one group is possible. This does not mean that every group must be checked: every resolvent can cancel out many possible groups. For example, for degree five polynomials there is never need for a resolvent of formula_14: resolvents for formula_15 and formula_16 give desired information.
One way is to begin from maximal (transitive) subgroups until the right one is found and then continue with maximal subgroups of that.