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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 yet a fundamental tool to compute Galois groups. The simplest examples of resolvents are

• ${\displaystyle X^{2}-\Delta }$ where ${\displaystyle \Delta }$ is the discriminant, which is a resolvent for the alternating group. In the case of a cubic equation this resolvent is sometimes called resolvent quadratic; its roots appear explicitly in the formulas for the roots of a cubic equation.
• The cubic resolvent of a quartic equation which is a resolvent for the dihedral group of 8 elements.
• The Cayley resolvent is a resolvent for the maximal resoluble Galois group in degree five. It is a polynomial of degree 6.

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 n be a positive integer which will be the degree of the equation that we will consider, and ${\displaystyle (X_{1},\ldots ,X_{n})}$ an ordered list of indeterminates. This defines the generic polynomial of degree n

${\displaystyle F(X)=X^{n}+\sum _{i=1}^{n}(-1)^{i}E_{i}X^{n-i}=\prod _{i=1}^{n}(X-X_{i}),}$

where E_i is the ith elementary symmetric polynomial.

The symmetric group S_n acts on the X_i by permuting them, and this induces an action on the polynomials in the X_i. The orbit of a given polynomial under this action is generally the whole group S_n, but some polynomials have a smaller orbit. For example, the orbit of an elementary symmetric polynomial is reduced to itself. If the orbit is not the whole symmetric group, the polynomial is fixed by some subgroup G; it is said an invariant of G. Conversely, given a subgroup G of S_n, an invariant of G is a resolvent invariant for G if it is not an invariant of any larger subgroup of S_n.

Finding resolvent invariants for a given group is relatively easy: for example one may choose a monomial and consider the sum of the monomials in this orbit. In the case of the subgroup D₄ of order 8 of S₄, the monomial ${\displaystyle X_{1}X_{2}}$ gives, for one of the possible actions of D₄ the invariant ${\displaystyle X_{1}X_{2}+X_{3}X4}$, which is a resolvent invariant for this group, used to define the cubic resolvent of the quartic equation.

If P is a resolvent invariant for a group G of index g, then its orbit under S_n has an order g. Let ${\displaystyle P_{1},\ldots ,P_{g}}$ be the elements of this orbit. Then the polynomial

${\displaystyle R_{G}=\prod _{i=1}^{g}(Y-P_{i})}$

is invariant under S_n. Thus, when expanded, its coefficients are polynomials in the X_i 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, R_G is an irreducible polynomial in Y whose coefficients are polynomial in the coefficients of F. Having the resolvent invariant as a root, it is called a resolvent (sometimes resolvent equation).

Let us consider now an irreducible polynomial

${\displaystyle f(X)=X^{n}+\sum _{i=1}^{n}a_{i}X^{n-i}=\prod _{i=1}^{n}(X-x_{i}),}$

with coefficients in a given field K (typically the field of rationals) and roots x_i in an algebraically closed field extension. Substituting the X_i by the x_i and the coefficients of F by those of f in what precedes, we get a polynomial ${\displaystyle R_{G}^{(f)}(Y)}$, also called resolvent or specialized resolvent in case of ambiguity). If the Galois group of f is contained in G, the specialization of the resolvent invariant is invariant by G and is thus a root of ${\displaystyle R_{G}^{(f)}(Y)}$ that belongs to K (is rational on K). Conversely, if ${\displaystyle R_{G}^{(f)}(Y)}$ has a rational root, which is not a multiple root, the Galois group of f is contained in G.

References

• Dickson, Leonard E. (1959). Algebraic Theories. New York: Dover Publications Inc. p. ix+276. ISBN 0-486-49573-6. {{cite book}}: Cite has empty unknown parameter: |1= (help)
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