Namespaces
Variants
Actions

Difference between revisions of "Resolvent"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (link)
m (tex encoded by computer)
Line 1: Line 1:
A resolvent of an algebraic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r0816001.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r0816002.png" /> is an algebraic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r0816003.png" />, with coefficients that rationally depend on the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r0816004.png" />, such that, if the roots of this equation are known, the roots of the given equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r0816005.png" /> can be found by solving simpler equations of degrees not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r0816006.png" />. A rational expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r0816007.png" /> itself is sometimes called a resolvent.
+
<!--
 +
r0816001.png
 +
$#A+1 = 102 n = 0
 +
$#C+1 = 102 : ~/encyclopedia/old_files/data/R081/R.0801600 Resolvent
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r0816008.png" /> be a [[separable polynomial]] over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r0816009.png" /> with [[Galois group|Galois group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160010.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160011.png" /> be a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160012.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160013.png" /> be a rational expression in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160014.png" /> that remains invariant under all permutations of the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160015.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160016.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160017.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160018.png" /> is a root of some equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160019.png" /> with coefficients from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160020.png" />, the Galois group of which is a proper quotient group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160021.png" />. Thus, solving the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160022.png" /> reduces to solving the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160023.png" /> and solving the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160024.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160025.png" />.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
For example, in order to solve an equation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160026.png" />:
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160027.png" /></td> </tr></table>
+
Let  $  f( x) $
 +
be a [[separable polynomial]] over a field  $  k $
 +
with [[Galois group|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} ) $.
  
(every equation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160028.png" /> is reducible to this form), the following cubic resolvent is used:
+
For example, in order to solve an equation of degree $  4 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160029.png" /></td> </tr></table>
+
$$
 +
x  ^ {4} + px  ^ {2} + qx + r  = 0
 +
$$
  
Its roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160030.png" /> are related to the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160031.png" /> by the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160034.png" />. The roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160035.png" /> are determined by the [[Cardano formula|Cardano formula]], which also makes it possible to determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160036.png" />.
+
(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|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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160037.png" /> be an equation over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160038.png" /> with a cyclic Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160039.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160040.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160041.png" /> contain a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160042.png" />-th root of unity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160043.png" />. For an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160044.png" /> which belongs to the splitting field of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160045.png" /> (cf. [[Splitting field of a polynomial|Splitting field of a polynomial]]), and for a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160046.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160047.png" /> into the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160048.png" />-th roots of unity, Lagrange's resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160049.png" /> is defined by the formula:
+
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|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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\rho ( \chi , \alpha )  = \sum _ {\sigma \in G } \chi ( \sigma )  ^ {-} 1
 +
\sigma ( \alpha ).
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160051.png" /> be one of the roots of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160052.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160053.png" /> run through the characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160054.png" />. Then for the system of linear equations (*) the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160055.png" /> can be determined if the Lagrange resolvents are known for all characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160057.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160058.png" /> the relation
+
For $  \tau \in G $
 +
the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160059.png" /></td> </tr></table>
+
$$
 +
\tau \rho ( \chi , \alpha )  = \
 +
\xi ( \tau ) \rho ( \chi , \alpha )
 +
$$
  
is fulfilled, showing that the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160061.png" />, for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160062.png" />, are invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160063.png" /> and are therefore uniquely defined rational expressions in the coefficients of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160064.png" /> and the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160065.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160066.png" /> generates the group of characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160067.png" />, then the following equalities hold: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160069.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160070.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160071.png" /> that is irreducible over a given field (see [[Galois theory|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160072.png" />, is called a Galois resolvent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160073.png" />.
+
Any algebraic equation $  y( x) = 0 $
 +
that is irreducible over a given field (see [[Galois theory|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====
 
====References====
Line 36: Line 121:
 
The resolvent (resolvent kernel) of an [[Integral equation|integral equation]]
 
The resolvent (resolvent kernel) of an [[Integral equation|integral equation]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160074.png" /></td> <td valign="top" style="width:5%;text-align:right;">(**)</td></tr></table>
+
$$ \tag{** }
 +
\phi ( s) + \lambda \int\limits _ { a } ^ { b }  K( s, t) \phi ( t)  dt  = f( s)
 +
$$
 +
 
 +
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:
  
is understood to be a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160075.png" /> of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160076.png" /> and the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160077.png" /> with the aid of which the solution of equation (**) can be represented in the form:
+
$$
 +
f( s) + \lambda \int\limits _ { a } ^ { b }  \Gamma ( s, t, \lambda ) f( t) dt ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160078.png" /></td> </tr></table>
+
provided  $  \lambda $
 +
is not an eigenvalue of (**). For example, for the kernel  $  K( s, t) = s + t $
 +
the resolvent is the function
  
provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160079.png" /> is not an eigenvalue of (**). For example, for the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160080.png" /> the resolvent is the function
+
$$
 +
\Gamma ( s, t, \lambda ) = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160081.png" /></td> </tr></table>
+
\frac{s+ t - (( s+ t) / 2 - st - 1 / 3 ) \lambda }{1 - \lambda - {
 +
\lambda  ^ {2} } / 12 }
 +
.
 +
$$
  
 
''BSE-3''
 
''BSE-3''
  
The resolvent of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160082.png" /> is an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160083.png" /> inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160084.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160085.png" /> is a closed linear operator defined on a dense set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160086.png" /> of a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160087.png" /> with values in the same space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160088.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160089.png" /> is a continuous linear operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160090.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160091.png" /> for which the resolvent exists are called regular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160092.png" />, and the collection of all regular points is the resolvent set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160093.png" /> of this operator. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160094.png" /> is open and on each of its connected components the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160095.png" /> is an analytic function of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160096.png" />.
+
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:
 
Properties of a resolvent are:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160097.png" /> for any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160098.png" />;
+
1) $  R _  \lambda  - R _  \mu  = ( \lambda - \mu ) R _  \lambda  R _  \mu  $
 +
for any two points $  \lambda , \mu \in \rho ( A) $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r08160099.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r081600100.png" />;
+
2) $  R _  \lambda  x = 0 $
 +
implies $  x = 0 $;
  
3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r081600101.png" /> is a Hilbert space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081600/r081600102.png" />.
+
3) if $  X $
 +
is a Hilbert space, then $  R _ {\overline \lambda \; }  = R _  \lambda  ^ {*} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis in normed spaces" , Pergamon  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis in normed spaces" , Pergamon  (1964)  (Translated from Russian)</TD></TR></table>

Revision as of 08:11, 6 June 2020


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=36096
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article