Resolvent

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)