Resolvent (Galois theory)

Let n be a positive integer, which will be the degree of the equation that we will consider, and (X₁, ..., 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 i^th 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 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 S_n. 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 S_n, an invariant of G is a resolvent invariant for G if it is not an invariant of any bigger subgroup of S_n.[1]

Finding invariants for a given subgroup G of S_n is relatively easy; one can sum the orbit of a monomial under the action of S_n. 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). The monomial X₁X₂ gives the invariant 2(X₁X₂ + X₃X₄). It is not a resolvent invariant for G, as being invariant by (12), in fact, it is a resolvent invariant for the dihedral subgroup ⟨(12), (1324)⟩, and is used to define the resolvent cubic of the quartic equation.

If P is a resolvent invariant for a group G of index m, then its orbit under S_n has order m. Let P₁, ..., P_m be the elements of this orbit. Then the polynomial

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

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).

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.

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 Lagrange resolvent may refer to the linear polynomial

${\displaystyle \sum _{i=0}^{n-1}X_{i}\omega ^{i}}$

where ${\displaystyle \omega }$ 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 H of S_n, having the property that, if a relative resolvent for a subgroup G of H has a rational simple root and the Galois group of f is contained in H, then the Galois group of f is contained in G. In this context, a usual resolvent is called an absolute resolvent.

The Galois group of a polynomial of degree ${\displaystyle n}$ is ${\displaystyle S_{n}}$ 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 ${\displaystyle S_{n}}$ 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 ${\displaystyle D_{5}}$: resolvents for ${\displaystyle A_{5}}$ and ${\displaystyle M_{20}}$ 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.

• Dickson, Leonard E. (1959). Algebraic Theories. New York: Dover Publications Inc. p. ix+276. ISBN 0-486-49573-6.
• Girstmair, K. (1983). "On the computation of resolvents and Galois groups". Manuscripta Mathematica. 43 (2–3): 289–307. doi:10.1007/BF01165834. S2CID 123752910.