Resolvent (Galois theory)

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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 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[edit]

Let n be a positive integer, which will be the degree of the equation that we will consider, and (X_1, \ldots, X_n) an ordered list of indeterminates. This defines the generic polynomial of degree n

F(X)=X^n+\sum_{i=1}^n (-1)^i E_i X^{n-i} = \prod_{i=1}^n (X-X_i),

where Ei is the ith elementary 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 orbit of a given polynomial under this action is generally the whole group Sn, 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 Sn, an invariant of G is a resolvent invariant for G if it is not an invariant of any larger subgroup of Sn.

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 X_1 X_2 gives, for one of the possible actions of D4 the invariant X_1 X_2 +X_3 X_4, 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 Sn has an order g. Let P_1, \ldots, P_g be the elements of this orbit. Then the polynomial

R_G=\prod_{i=1}^g (Y-P_i)

is invariant under Sn. Thus, when expanded, its coefficients are polynomials in the Xi 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, RG 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

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 xi in an algebraically closed field extension. Substituting the Xi by the xi and the coefficients of F by those of f in what precedes, we get a polynomial 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 R_G^{(f)}(Y) that belongs to K (is rational on K). Conversely, if R_G^{(f)}(Y) has a rational root, which is not a multiple root, the Galois group of f is contained in G.

Terminology[edit]

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
\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 H of Sn, 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.

Resolvent method[edit]

The Galois group of a polynomial of degree n is S_n or a proper subgroup of that. If a polynomial is 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[edit]