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

• $X^2-\mathrm{\Delta }$ where $\mathrm{\Delta }$ is the discriminant, which is a resolvent for the alternating group. In the case of a cubic equation, this resolvent is sometimes called the quadratic resolvent; 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 solvable 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 solvable group, because the Galois group of the equation over the field generated by this root is solvable.

Definition

Let Template:Mvar be a positive integer, which will be the degree of the equation that we will consider, and Template:Math an ordered list of indeterminates. According to Vieta's formulas this defines the generic monic polynomial of degree Template:Mvar $$F(X)=X^n+\sum _{i=1}^n(-1{)}^i{E}_i{X}^{n-i}=\prod _{i=1}^n(X-X_i),$$ where Template:Math is the Template:Math th elementary symmetric polynomial.

The symmetric group Template:Math acts on the Template:Math by permuting them, and this induces an action on the polynomials in the Template:Math. 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 Template:Math. If the stabilizer is non-trivial, the polynomial is fixed by some non-trivial subgroup Template:Mvar; it is said to be an invariant of Template:Mvar. Conversely, given a subgroup Template:Mvar of Template:Math, an invariant of Template:Mvar is a resolvent invariant for Template:Mvar if it is not an invariant of any bigger subgroup of Template:Math.

Finding invariants for a given subgroup Template:Mvar of Template:Math is relatively easy; one can sum the orbit of a monomial under the action of Template:Math. However, it may occur that the resulting polynomial is an invariant for a larger group. For example, consider the case of the subgroup Template:Mvar of Template:Math of order 4, consisting of Template:Math, Template:Math, Template:Math and the identity (for the notation, see Permutation group). The monomial Template:Math gives the invariant Template:Math. It is not a resolvent invariant for Template:Mvar, because being invariant by Template:Math, it is in fact a resolvent invariant for the larger dihedral subgroup Template:Math: Template:Math, and is used to define the resolvent cubic of the quartic equation.

If Template:Mvar is a resolvent invariant for a group Template:Mvar of index Template:Mvar inside Template:Math, then its orbit under Template:Math has order Template:Mvar. Let Template:Math be the elements of this orbit. Then the polynomial

$R_G=\prod _{i=1}^m(Y-P_i)$

is invariant under Template:Math. Thus, when expanded, its coefficients are polynomials in the Template:Math 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, Template:Math is an irreducible polynomial in Template:Mvar whose coefficients are polynomial in the coefficients of Template:Mvar. Having the resolvent invariant as a root, it is called a resolvent (sometimes resolvent equation).

Consider now an irreducible polynomial

$f(X)=X^n+\sum _{i=1}^n{a}_i{X}^{n-i}=\prod _{i=1}^n(X-x_i),$

with coefficients in a given field Template:Mvar (typically the field of rationals) and roots Template:Math in an algebraically closed field extension. Substituting the Template:Math by the Template:Math and the coefficients of Template:Mvar by those of Template:Mvar in the above, we get a polynomial $R_G^{(f)}(Y)$, also called resolvent or specialized resolvent in case of ambiguity). If the Galois group of Template:Mvar is contained in Template:Mvar, the specialization of the resolvent invariant is invariant by Template:Mvar and is thus a root of $R_G^{(f)}(Y)$ that belongs to Template:Mvar (is rational on Template:Mvar). Conversely, if $R_G^{(f)}(Y)$ has a rational root, which is not a multiple root, the Galois group of Template:Mvar is contained in Template:Mvar.

Terminology

There are some variants in the terminology.

• Depending on the authors or on the context, resolvent may refer to resolvent invariant instead of to resolvent equation.
• A Galois resolvent is a resolvent such that the resolvent invariant is linear in the roots.
• The Template:Vanchor may refer to the linear polynomial $$\sum _{i=0}^{n-1}{X}_i{\omega }^i$$ where $\omega$ is a primitive nth root of unity. It is the resolvent invariant of a Galois resolvent for the identity group.
• A relative resolvent is defined similarly as a resolvent, but considering only the action of the elements of a given subgroup Template:Mvar of Template:Math, having the property that, if a relative resolvent for a subgroup Template:Mvar of Template:Mvar has a rational simple root and the Galois group of Template:Mvar is contained in Template:Mvar, then the Galois group of Template:Mvar is contained in Template:Mvar. In this context, a usual resolvent is called an absolute resolvent.

Resolvent method

The Galois group of a polynomial of degree $n$ is $S_n$ or a proper subgroup of it. If a polynomial is separable and irreducible, then the corresponding Galois group is a transitive subgroup.

Transitive subgroups of $S_n$ 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 $D_5$: resolvents for $A_5$ and $M_{20}$ 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.

References

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