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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 number|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 (polynomial)|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 group]]s. The simplest examples of resolvents are
* <math>X^2-\Delta</math> where <math>\Delta</math> 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 [[resolvent cubic|cubic resolvent]] of a [[quartic function|quartic equation]] which is a resolvent for the [[dihedral group]] of 8 elements.
* The [[Quintic function#Solvable quintics|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 {{mvar|''n''}} be a positive integer which will be the degree of the equation that we will consider, and <math>(X_1, \ldots, X_n)</math> an ordered list of [[indeterminate (variable)|indeterminates]]. This defines the ''generic polynomial'' of degree {{mvar|''n''}}
:<math>F(X)=X^n+\sum_{i=1}^n (-1)^i E_i X^{n-i} = \prod_{i=1}^n (X-X_i),</math>
where {{math|''E''''i''}} is the ''i''th [[elementary symmetric polynomial]].

The [[symmetric group]] {{math|''S''''n''}} acts on the {{math|''X''''i''}} by permuting them, and this induces an action on the polynomials in the {{math|''X''''i''}}. The [[orbit (group theory)|orbit]] of a given polynomial under this action is generally the whole group {{math|''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 {{mvar|''G''}}; it is said an ''invariant'' of {{mvar|''G''}}. Conversely, given a subgroup {{mvar|''G''}} of {{math|''S''''n''}}, an invariant of {{mvar|''G''}} is a '''resolvent invariant''' for {{mvar|''G''}} if it is not an invariant of any larger subgroup of {{math|''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 {{math|''D''''4''}} of order 8 of {{math|''S''''4''}}, the monomial <math>X_1 X_2</math> gives, for one of the possible actions of {{math|''D''''4''}} the invariant <math>X_1 X_2 +X_3 X_4</math>, which is a resolvent invariant for this group, used to define the [[cubic resolvent]] of the [[quartic equation]].

If {{mvar|''P''}} is a resolvent invariant for a group {{mvar|''G''}} of [[index (group theory)|index]] {{mvar|''g''}}, then its orbit under {{math|''S''''n''}} has an order {{mvar|''g''}}. Let <math>P_1, \ldots, P_g</math> be the elements of this orbit. Then the polynomial
:<math>R_G=\prod_{i=1}^g (Y-P_i)</math>
is invariant under {{math|''S''''n''}}. Thus, when expanded, its coefficients are polynomials in the {{math|''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, {{math|''R''''G''}} is an [[irreducible polynomial]] in {{mvar|''Y''}} whose coefficients are polynomial in the coefficients of {{mvar|''F''}}. Having the resolvent invariant as a root, it is called a '''resolvent''' (sometimes '''resolvent equation''').

Let us consider now an irreducible polynomial
:<math>f(X)=X^n+\sum_{i=1}^n a_i X^{n-i} = \prod_{i=1}^n (X-x_i),</math>
with coefficients in a given field {{mvar|''K''}} (typically the [[field of rationals]]) and roots {{math|''x''''i''}} in an [[algebraically closed field]] extension. Substituting the {{math|''X''''i''}} by the {{math|''x''''i''}} and the coefficients of {{mvar|''F''}} by those of {{mvar|''f''}} in what precedes, we get a polynomial <math>R_G^{(f)}(Y)</math>, also called ''resolvent'' or ''specialized resolvent'' in case of ambiguity). If the [[Galois group]] of {{mvar|''f''}} is contained in {{mvar|''G''}}, the specialization of the resolvent invariant is invariant by {{mvar|''G''}} and is thus a root of <math>R_G^{(f)}(Y)</math> that belongs to {{mvar|''K''}} (is rational on {{mvar|''K''}}). Conversely, if <math>R_G^{(f)}(Y)</math> has a rational root, which is not a multiple root, the Galois group of {{mvar|''f''}} is contained in {{mvar|''G''}}.

==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 '''Lagrange resolvent''' may refer to the linear polynomial
::<math>\sum_{i=0}^{n-1} X_i \omega^i</math>
:where <math>\omega</math> is a [[primitive nth root of unity|primitive ''n''th 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 {{mvar|''H''}} of {{math|''S''''n''}}, having the property that, if a relative resolvent for a subgroup {{mvar|''G''}} of {{mvar|''H''}} has a rational simple root and the Galois group of {{mvar|''f''}} is contained in {{mvar|''H''}}, then the Galois group of {{mvar|''f''}} is contained in {{mvar|''G''}}. In this context, a usual resolvent is called an '''absolute resolvent'''.

==Resolvent method==

The Galois group of a polynomial of degree <math>n</math> is <math>S_n</math> or a proper subgroup of that. If a polynomial is irreducible, then the corresponding Galois group is a transitive subgroup.

Transitive subgroups of <math>S_n</math> 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 <math>D_5</math>: resolvents for <math>A_5</math> and <math>M_{20}</math> 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 ==
* {{cite book |title=Algebraic Theories |first= Leonard E.|last=Dickson |authorlink=Leonard Eugene Dickson |publisher= Dover Publications Inc|location= New York|year= 1959|isbn=0-486-49573-6 |pages= |page= ix+276}}
* {{cite doi|10.1007/BF01165834|noedit}}

[[Category:Field theory]]
[[Category:Group theory]]
[[Category:Galois theory]]
[[Category:Equations]]

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en>Donner60: Reverted 1 edit by 67.251.120.21 (talk) identified as vandalism to last revision by BG19bot. (TW)