In Galois theory, a discipline within the field of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...

, a resolvent for a permutation group ''G'' is a polynomial whose

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...

s depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a

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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 Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaÉvariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are *

X^2-\Delta

where

\Delta

is the discriminant, which is a resolvent for the

alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...

. In the case of a

cubic equation In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...

, 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 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

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and of a root of a resolvent for a resoluble group, because, the Galois group of the equation over the

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

generated by this root is resoluble.

Definition

Let be a positive integer, which will be the degree of the equation that we will consider, and an ordered list of indeterminates. This defines the ''generic polynomial'' of degree

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

where is the th

elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...

. The symmetric group

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on the by permuting them, and this induces an action on the polynomials in the . 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

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. If the stabilizer is non-trivial, the polynomial is fixed by some non-trivial

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...

; it is said to be an ''invariant'' of . Conversely, given a subgroup of , an invariant of is a resolvent invariant for if it is not an invariant of any bigger subgroup of .http://www.alexhealy.net/papers/math250a.pdf Finding invariants for a given subgroup of is relatively easy; one can sum the orbit of a

monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...

under the action of . However, it may occur that the resulting polynomial is an invariant for a larger group. For example, consider the case of the subgroup of of order 4, consisting of , , and the identity (for the notation, see Permutation group). The monomial gives the invariant . It is not a resolvent invariant for , because being invariant by , it is in fact a resolvent invariant for the larger dihedral subgroup : , and is used to define the resolvent cubic of the quartic equation. If is a resolvent invariant for a group of

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inside , then its orbit under has order . Let be the elements of this orbit. Then the polynomial :

R_G=\prod_^m (Y-P_i)

is invariant under . Thus, when expanded, its coefficients are polynomials in the 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, is an irreducible polynomial in whose coefficients are polynomial in the coefficients of . 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_^n a_i X^ = \prod_^n (X-x_i),

with coefficients in a given field (typically the

field of rationals In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...

) and roots in an algebraically closed field extension. Substituting the by the and the coefficients of by those of in the above, we get a polynomial

R_G^(Y)

, also called ''resolvent'' or ''specialized resolvent'' in case of ambiguity). If the Galois group of is contained in , the specialization of the resolvent invariant is invariant by and is thus a root of

R_G^(Y)

that belongs to (is rational on ). Conversely, if

R_G^(Y)

has a rational root, which is not a multiple root, the Galois group of is contained in .

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 may refer to the linear polynomial

\sum_^ X_i \omega^i

where

\omega

is a 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 of , having the property that, if a relative resolvent for a subgroup of has a rational simple root and the Galois group of is contained in , then the Galois group of is contained in . In this context, a usual resolvent is called an absolute resolvent.

Resolvent method

The Galois group of a polynomial of degree

n

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_

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 journal , last1 = Girstmair , first1 = K. , title = On the computation of resolvents and Galois groups , doi = 10.1007/BF01165834 , journal = Manuscripta Mathematica , volume = 43 , issue = 2–3 , pages = 289–307 , year = 1983 , s2cid = 123752910 Group theory Galois theory Equations