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Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
, 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 In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to i ...
''G'' is a
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
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
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...
the
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
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 Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical ...
is true if the rational root is a
simple root Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
. Resolvents were introduced by
Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaÉvariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
. Nowadays they are still a fundamental tool to compute
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
s. The simplest examples of resolvents are * X^2-\Delta where \Delta is the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
, 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 In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomi ...
, which is a resolvent for the
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
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 In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multipl ...
, 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 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 An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
, 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 In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \ ...
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
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
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. 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 In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
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 In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to i ...
). 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 In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four: :P(x)=x^4+a_3x^3+a_2x^2+a_1x+a_0. In each case: * The coefficients of the resolvent cubic can be obta ...
of the
quartic equation In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomi ...
. If is a resolvent invariant for a group of
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
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 mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted ...
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 In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
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 is S_n or a
proper 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 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