Resolvent

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A resolvent of an algebraic equation $f( x) = 0$ of degree $n$ is an algebraic equation $g( y) = 0$, with coefficients that rationally depend on the coefficients of $f( x)$, such that, if the roots of this equation are known, the roots of the given equation $f( x) = 0$ can be found by solving simpler equations of degrees not exceeding $n$. A rational expression $y = y( x _ {1} \dots x _ {n} )$ itself is sometimes called a resolvent.

Let $f( x)$ be a separable polynomial over a field $k$ with Galois group $G$, and let $H$ be a normal subgroup of $G$. Let $y = y( x _ {1} \dots x _ {n} )$ be a rational expression in $x _ {1} \dots x _ {n}$ that remains invariant under all permutations of the roots $x _ {1} \dots x _ {n}$ belonging to $H$, and let $y \notin k$. Then $y$ is a root of some equation $g( y) = 0$ with coefficients from $k$, the Galois group of which is a proper quotient group of $G$. Thus, solving the equation $f( x) = 0$ reduces to solving the equation $g( y) = 0$ and solving the equation $f( x) = 0$ over the field $k( y _ {1} \dots y _ {s} )$.

For example, in order to solve an equation of degree $4$:

$$x ^ {4} + px ^ {2} + qx + r = 0$$

(every equation of degree $4$ is reducible to this form), the following cubic resolvent is used:

$$y ^ {3} - 2py ^ {2} + ( p ^ {2} - 4r ) y + q ^ {2} = 0.$$

Its roots $y _ {1} , y _ {2} , y _ {3}$ are related to the roots $x _ {1} , x _ {2} , x _ {3} , x _ {4}$ by the relations $y _ {1} = ( x _ {1} + x _ {2} )( x _ {3} + x _ {4} )$, $y _ {2} = ( x _ {1} + x _ {3} )( x _ {2} + x _ {4} )$, $y _ {3} = ( x _ {1} + x _ {4} )( x _ {2} + x _ {3} )$. The roots $y _ {1} , y _ {2} , y _ {3}$ are determined by the Cardano formula, which also makes it possible to determine $x _ {1} , x _ {2} , x _ {3} , x _ {4}$.

Successive application of the resolvent method permits one to solve any equation with a solvable Galois group by reduction to solving a chain of equations with cyclic Galois groups. Lagrange's resolvent is used in solving the latter.

Let $f( x) = 0$ be an equation over a field $k$ with a cyclic Galois group $G$ of order $n$, and let $k$ contain a primitive $n$- th root of unity $\zeta _ {n}$. For an element $\alpha$ which belongs to the splitting field of the polynomial $f( x)$( cf. Splitting field of a polynomial), and for a character $\chi$ from $G$ into the group of $n$- th roots of unity, Lagrange's resolvent $\rho ( \chi , \alpha )$ is defined by the formula:

$$\tag{* } \rho ( \chi , \alpha ) = \sum _ {\sigma \in G } \chi ( \sigma ) ^ {-} 1 \sigma ( \alpha ).$$

Let $\alpha = x _ {1}$ be one of the roots of the polynomial $f( x)$ and let $\chi$ run through the characters of $G$. Then for the system of linear equations (*) the roots $x _ {1} \dots x _ {n}$ can be determined if the Lagrange resolvents are known for all characters $\chi$ of $G$.

For $\tau \in G$ the relation

$$\tau \rho ( \chi , \alpha ) = \ \xi ( \tau ) \rho ( \chi , \alpha )$$

is fulfilled, showing that the elements $a = \rho ( \chi , \alpha ) ^ {n}$ and $b _ {i} = \rho ( \chi , \alpha ) ^ {-} i \rho ( \chi ^ {i} , \alpha )$, for any integer $i$, are invariant under $G$ and are therefore uniquely defined rational expressions in the coefficients of the polynomial $f( x)$ and the root $\zeta _ {n}$. If $\chi$ generates the group of characters of $G$, then the following equalities hold: $\rho ( \chi , \alpha ) = a ^ {1/n}$ and $\rho ( \chi ^ \prime , \alpha ) = b _ {i} \rho ( \chi , \alpha ) ^ {i}$ for $\chi ^ \prime = \chi ^ {i}$.

Any algebraic equation $y( x) = 0$ that is irreducible over a given field (see Galois theory) and that is such that as a result of the adjunction of one of its roots to this field a field is obtained that contains all roots of the equation $f( x) = 0$, is called a Galois resolvent of $f( x)$.

References

 [1] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)

L.V. Kuz'min

The resolvent (resolvent kernel) of an integral equation

$$\tag{** } \phi ( s) + \lambda \int\limits _ { a } ^ { b } K( s, t) \phi ( t) dt = f( s)$$

is understood to be a function $\Gamma ( s, t, \lambda )$ of the variables $s, t$ and the parameter $\lambda$ with the aid of which the solution of equation (**) can be represented in the form:

$$f( s) + \lambda \int\limits _ { a } ^ { b } \Gamma ( s, t, \lambda ) f( t) dt ,$$

provided $\lambda$ is not an eigenvalue of (**). For example, for the kernel $K( s, t) = s + t$ the resolvent is the function

$$\Gamma ( s, t, \lambda ) = \ \frac{s+ t - (( s+ t) / 2 - st - 1 / 3 ) \lambda }{1 - \lambda - { \lambda ^ {2} } / 12 } .$$

BSE-3

The resolvent of an operator $A$ is an operator $R _ \lambda$ inverse to $T _ \lambda = A - \lambda I$. Here $A$ is a closed linear operator defined on a dense set $D _ {A}$ of a Banach space $X$ with values in the same space and $\lambda$ is such that $T _ \lambda ^ {-} 1$ is a continuous linear operator on $X$. The points $\lambda$ for which the resolvent exists are called regular points of $A$, and the collection of all regular points is the resolvent set $\rho ( A)$ of this operator. The set $\rho ( A)$ is open and on each of its connected components the operator $R _ \lambda$ is an analytic function of the parameter $\lambda$.

Properties of a resolvent are:

1) $R _ \lambda - R _ \mu = ( \lambda - \mu ) R _ \lambda R _ \mu$ for any two points $\lambda , \mu \in \rho ( A)$;

2) $R _ \lambda x = 0$ implies $x = 0$;

3) if $X$ is a Hilbert space, then $R _ {\overline \lambda \; } = R _ \lambda ^ {*}$.

References

 [1] K. Yosida, "Functional analysis" , Springer (1980) [2] N.I. [N.I. Akhiezer] Achieser, I.M. [I.M. Glaz'man] Glasman, "Theorie der linearen Operatoren im Hilbert Raum" , Akademie Verlag (1954) (Translated from Russian) [3] L.V. Kantorovich, G.P. Akilov, "Functional analysis in normed spaces" , Pergamon (1964) (Translated from Russian)
How to Cite This Entry:
Resolvent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resolvent&oldid=48530
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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