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A [[Differential field|differential field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e0369602.png" /> with a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e0369603.png" /> of differentiations such that the set of restrictions of the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e0369604.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e0369605.png" /> coincides with the set of differentiations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e0369606.png" />. In turn, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e0369607.png" /> is a differential subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e0369608.png" />.
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'' $  F _{0} $''
  
The intersection of any set of differential subfields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e0369609.png" /> is again a differential subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696010.png" />. For any set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696011.png" /> there is a smallest differential subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696012.png" /> containing all the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696014.png" />; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696015.png" /> and is called the extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696016.png" /> generated by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696017.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696018.png" /> is called a set, or family, of generators of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696019.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696020.png" />). The extension is said to be finitely generated if it has a finite set of generators, and is called simply generated if the set of generators consists of one element. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696022.png" /> are two differential subfields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696023.png" />, then the subfield
 
  
<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/e/e036/e036960/e03696024.png" /></td> </tr></table>
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A [[Differential field|differential field]]  $  F \supset F _{0} $
 +
with a set  $  \Delta $
 +
of differentiations such that the set of restrictions of the elements of  $  \Delta $
 +
to  $  F _{0} $
 +
coincides with the set of differentiations on  $  F _{0} $.  
 +
In turn,  $  F _{0} $
 +
is a differential subfield of  $  F $.
  
is a differential subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696025.png" />, called the join of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696027.png" />.
 
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696028.png" /> be the free commutative semi-group generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696029.png" /> (its elements are called differential operators). A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696030.png" /> of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696031.png" /> is said to be differentially algebraically dependent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696032.png" /> if the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696033.png" /> is algebraically dependent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696034.png" />. In the opposite case, the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696035.png" /> is called differentially algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696036.png" />, or a family of differential indeterminates over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696037.png" />. One says that the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696038.png" /> are differentially separably dependent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696039.png" /> if the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696040.png" /> is separably dependent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696041.png" />. In the opposite case the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696042.png" /> is called differentially separably independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696043.png" />.
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The intersection of any set of differential subfields of  $  F $
 +
is again a differential subfield of  $  F $.  
 +
For any set of elements $  \Sigma \subset F $
 +
there is a smallest differential subfield of $  F $
 +
containing all the elements of $  \Sigma $
 +
and  $  F _{0} $;
 +
it is denoted by  $  F _{0} \langle  \Sigma \rangle $
 +
and is called the extension of the field  $  F _{0} $
 +
generated by the set  $  \Sigma $(
 +
and  $  \Sigma $
 +
is called a set, or family, of generators of the extension  $  F _{0} \langle  \Sigma \rangle $
 +
over $  F _{0} $).  
 +
The extension is said to be finitely generated if it has a finite set of generators, and is called simply generated if the set of generators consists of one element. If  $  F _{1} $
 +
and  $  F _{2} $
 +
are two differential subfields of  $  F $,
 +
then the subfield $$
 +
F _{1} F _{2}  =
 +
F _{1} \langle  F _{2} \rangle  =
 +
F _{1} (F _{2} )  =
 +
F _{2} (F _{1} )  =
 +
F _{2} \langle  F _{1} \rangle,
 +
$$
 +
is a differential subfield of  $  F $,
 +
called the join of the fields  $  F _{1} $
 +
and  $  F _{2} $.
  
An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696044.png" /> is called differentially algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696045.png" /> if every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696046.png" /> is differentially algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696047.png" />. Similarly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696048.png" /> is called differentially separable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696049.png" /> if every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696050.png" /> is differentially separable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696051.png" />. The theorem on the primitive element applies to differential extensions: If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696052.png" /> is independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696053.png" />, then every finitely-generated differentially-separable extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696055.png" /> is generated by one element.
 
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696056.png" /> be a given set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696057.png" /> be the polynomial algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696058.png" /> in the family of indeterminates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696059.png" />, with index set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696060.png" />. Any differentiation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696061.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696062.png" /> extends in a unique way to a differentiation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696063.png" /> sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696064.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696065.png" />. This differential ring is called the ring of differential polynomials in the differential indeterminates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696067.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696068.png" />. Its differential field of fractions (i.e. the field of fractions with extended differentiations) is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696069.png" />, and the elements of this field are called differential functions over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696070.png" /> in the differential indeterminates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696071.png" />. For ordinary differential fields an analogue of the Lüroth theorem holds: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696072.png" /> is an arbitrary differential extension of a differential field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696073.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696074.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696075.png" /> contains an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696076.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696077.png" />.
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Let $  \Theta $
 +
be the free commutative semi-group generated by  $  \Delta $(
 +
its elements are called differential operators). A family  $  ( \alpha _{i} ) _ {i \in I} $
 +
of elements of $  F $
 +
is said to be differentially algebraically dependent over  $  F _{0} \subset F $
 +
if the family  $  ( \theta \alpha _{i} ) _ {i \in I,\  \theta \in \Theta} $
 +
is algebraically dependent over  $  F _{0} $.  
 +
In the opposite case, the family  $  ( \alpha _{i} ) _ {i \in I} $
 +
is called differentially algebraically independent over  $  F _{0} $,  
 +
or a family of differential indeterminates over $  F _{0} $.  
 +
One says that the elements of  $  ( \alpha _{i} ) _ {i \in I} $
 +
are differentially separably dependent over  $  F _{0} $
 +
if the family  $  ( \theta \alpha _{i} ) _ {i \in I,\  \theta \in \Theta} $
 +
is separably dependent over  $  F _{0} $.  
 +
In the opposite case the family  $  ( \alpha _{i} ) _ {i \in I} $
 +
is called differentially separably independent over  $  F _{0} $.
  
For any differential field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696078.png" /> there is a separable semi-universal extension, i.e. an extension containing every finitely-generated separable extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696079.png" />. Moreover, there exists a separable universal extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696080.png" />, i.e. an extension which is semi-universal over every finitely-generated extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696081.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696082.png" />.
 
  
In the theory of differential fields there is no direct analogue of the notion of an algebraically closed field in ordinary field theory. To a certain extent, their role is played by constrainedly closed fields. The main property of such a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696083.png" /> is that any finite system of algebraic differential equations and inequalities with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696084.png" /> having a solution that is rational over some field extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696085.png" /> has a solution that is rational over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696086.png" />. A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696087.png" /> of elements of some extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696088.png" /> is called constrained over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696089.png" /> if there is a differential polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696090.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696092.png" /> for any non-generic differential specialization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696093.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696094.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696095.png" />. An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696096.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696097.png" /> is called constrained over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696098.png" /> if any finite set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696099.png" /> is constrained over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960100.png" />. This is equivalent to saying that an arbitrary element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960101.png" /> is constrained over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960102.png" />. A differential field having no non-trivial constrained extensions is called constrainedly closed. An example of such a field is the universal differential field of characteristic zero (the universal field extension of the field of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960103.png" />). Any differential field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960104.png" /> of characteristic zero has a constrained closure, i.e. a constrainedly closed extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960105.png" /> which is contained in any other constrainedly closed extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960106.png" />.
+
An extension $  F $
 +
is called differentially algebraic over $  F _{0} $
 +
if every element of  $  F $
 +
is differentially algebraic over $  F _{0} $.  
 +
Similarly,  $  F $
 +
is called differentially separable over $  F _{0} $
 +
if every element of $  F $
 +
is differentially separable over $  F _{0} $.  
 +
The theorem on the primitive element applies to differential extensions: If the set  $  \Theta $
 +
is independent over $  F _{0} $,
 +
then every finitely-generated differentially-separable extension $  F $
 +
of $  F _{0} $
 +
is generated by one element.
  
The notion of a [[Normal extension|normal extension]] in ordinary field theory carries over to differential algebra in various ways. In differential Galois theory, a fundamental role is played by strongly normal extensions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960107.png" /> be the fixed universal differential field of characteristic 0 with field of constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960108.png" />. All the differential fields encountered below are assumed to lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960109.png" /> and all isomorphisms are assumed to be differential isomorphisms, that is, they commute with the operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960110.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960112.png" /> be differential fields over which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960113.png" /> is universal. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960114.png" /> be the field of constants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960115.png" />. An isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960116.png" /> leaves invariant each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960118.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960119.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960120.png" />). A strongly normal extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960121.png" /> is a finitely-generated extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960122.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960123.png" /> such that every isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960124.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960125.png" /> is strong. Strongly normal extensions are constrained. The set of strong isomorphisms of a strongly normal extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960126.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960127.png" /> has the natural structure of an algebraic group, defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960128.png" /> (and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960129.png" />). This is the [[Galois differential group|Galois differential group]] of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960130.png" />. A special case of strongly normal extensions is given by the Picard–Vessiot extensions, i.e. extensions that preserve the field of constants and result from the adjunction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960131.png" /> of a basis for the solutions of some system of homogeneous linear differential equations with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960132.png" />. For extensions of this type, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960133.png" /> is an algebraic matrix group, i.e. an algebraic subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960134.png" /> for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960135.png" />.
+
Let  $  J $
 +
be a given set and let  $  F _{0} [(y _ {j \theta} ) _ {j \in J,\  \theta \in \Theta} ] $
 +
be the polynomial algebra over  $  F _{0} $
 +
in the family of indeterminates  $  (y _ {j \theta} ) _ {j \in J,\  \theta \in \Theta} $,  
 +
with index set  $  J \times \Theta $.
 +
Any differentiation  $  \delta \in \Delta $
 +
of  $  F _{0} $
 +
extends in a unique way to a differentiation of  $  F _{0} [(y _ {j \theta} ) _ {j \in J,\  \theta \in \Theta} ] $
 +
sending  $  y _ {j \theta} $
 +
to  $  y _ {j \delta \theta} $.  
 +
This differential ring is called the ring of differential polynomials in the differential indeterminates  $  y _{j} $,  
 +
$  j \in J $,  
 +
and is denoted by  $  F _{0} \{ (y _{j} ) _ {j \in J} \} $.  
 +
Its differential field of fractions (i.e. the field of fractions with extended differentiations) is denoted by  $  F _{0} \langle  (y _{j} ) _ {j \in J} \rangle $,
 +
and the elements of this field are called differential functions over $  F _{0} $
 +
in the differential indeterminates  $  (y _{j} ) _ {j \in J} $.  
 +
For ordinary differential fields an analogue of the Lüroth theorem holds: If  $  F $
 +
is an arbitrary differential extension of a differential field  $  F _{0} $
 +
contained in $  F _{0} \langle  u \rangle $,  
 +
then  $  F $
 +
contains an element  $  v $
 +
such that  $  F = F _{0} \langle  v \rangle $.
  
The Galois differential groups of some typical differential algebraic extensions have the following form.
 
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960136.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960137.png" /> satisfies the system of equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960139.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960141.png" />, and let the fields of constants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960143.png" /> coincide. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960144.png" /> is a Picard–Vessiot extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960145.png" /> and the Galois differential group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960146.png" /> is a subgroup of the multiplicative group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960147.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960148.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960149.png" /> is algebraic, it satisfies an equation of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960150.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960151.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960152.png" /> (the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960153.png" />-th roots of unity). In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960154.png" /> is called an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960155.png" /> by an exponent.
+
For any differential field  $  F $
 +
there is a separable semi-universal extension, i.e. an extension containing every finitely-generated separable extension of $  F $.  
 +
Moreover, there exists a separable universal extension  $  U $,  
 +
i.e. an extension which is semi-universal over every finitely-generated extension of  $  F $
 +
contained in  $  U $.
 +
 
 +
 
 +
In the theory of differential fields there is no direct analogue of the notion of an algebraically closed field in ordinary field theory. To a certain extent, their role is played by constrainedly closed fields. The main property of such a field  $  F $
 +
is that any finite system of algebraic differential equations and inequalities with coefficients in  $  F $
 +
having a solution that is rational over some field extension of $  F $
 +
has a solution that is rational over  $  F $.
 +
A family  $  \eta = (n _{j} ) _ {j \in J} $
 +
of elements of some extension of  $  F $
 +
is called constrained over  $  F $
 +
if there is a differential polynomial  $  c \in F \{ (y _{j} ) _ {j \in J} \} $
 +
such that  $  c ( \eta ) \neq 0 $
 +
and $  c ( \eta ^ \prime  ) = 0 $
 +
for any non-generic differential specialization  $  \eta ^ \prime  $
 +
of the point  $  \eta $
 +
over  $  F $.  
 +
An extension  $  {\mathcal G} $
 +
of  $  F $
 +
is called constrained over  $  F $
 +
if any finite set of elements  $  \eta _{1} \dots \eta _{n} \in {\mathcal G} $
 +
is constrained over  $  F $.  
 +
This is equivalent to saying that an arbitrary element of  $  {\mathcal G} $
 +
is constrained over  $  F $.  
 +
A differential field having no non-trivial constrained extensions is called constrainedly closed. An example of such a field is the universal differential field of characteristic zero (the universal field extension of the field of rational numbers  $  \mathbf Q $).  
 +
Any differential field  $  F $
 +
of characteristic zero has a constrained closure, i.e. a constrainedly closed extension of  $  F $
 +
which is contained in any other constrainedly closed extension of $  F $.
 +
 
 +
 
 +
The notion of a [[Normal extension|normal extension]] in ordinary field theory carries over to differential algebra in various ways. In differential Galois theory, a fundamental role is played by strongly normal extensions. Let  $  U $
 +
be the fixed universal differential field of characteristic 0 with field of constants  $  K $.  
 +
All the differential fields encountered below are assumed to lie in  $  U $
 +
and all isomorphisms are assumed to be differential isomorphisms, that is, they commute with the operators in  $  \Delta $.  
 +
Let  $  F $
 +
and  $  {\mathcal G} $
 +
be differential fields over which  $  U $
 +
is universal. Let  $  C $
 +
be the field of constants of $  {\mathcal G} $.  
 +
An isomorphism  $  \sigma $
 +
leaves invariant each element of  $  C $,
 +
$  \sigma {\mathcal G} \subset {\mathcal G} K $,
 +
and  $  {\mathcal G} \subset \sigma {\mathcal G} K $(
 +
that is, $  {\mathcal G} K = {\mathcal G} \sigma K $).  
 +
A strongly normal extension of  $  F $
 +
is a finitely-generated extension  $  {\mathcal G} $
 +
of  $  F $
 +
such that every isomorphism of  $  {\mathcal G} $
 +
over  $  F $
 +
is strong. Strongly normal extensions are constrained. The set of strong isomorphisms of a strongly normal extension  $  {\mathcal G} $
 +
over  $  F $
 +
has the natural structure of an algebraic group, defined over  $  K $(
 +
and denoted by  $  \mathop{\rm Gal}\nolimits ( {\mathcal G} /F \  ) $).  
 +
This is the [[Galois differential group|Galois differential group]] of the extension  $  {\mathcal G} /F $.  
 +
A special case of strongly normal extensions is given by the Picard–Vessiot extensions, i.e. extensions that preserve the field of constants and result from the adjunction to  $  F $
 +
of a basis for the solutions of some system of homogeneous linear differential equations with coefficients in  $  F $.  
 +
For extensions of this type, $  \mathop{\rm Gal}\nolimits ( {\mathcal G} /F \  ) $
 +
is an algebraic matrix group, i.e. an algebraic subgroup of the group  $  \mathop{\rm GL}\nolimits (n,\  K) $
 +
for some integer  $  n > 0 $.
 +
 
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960156.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960157.png" /> satisfies the system of equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960158.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960159.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960160.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960161.png" /> (such an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960162.png" /> is called primitive over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960163.png" />), and let the field of constants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960164.png" /> coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960165.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960166.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960167.png" /> is transcendental over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960168.png" />. The resulting extension is a Picard–Vessiot extension, and the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960169.png" /> is isomorphic to the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960170.png" />. Such extensions are called extensions by an integral.
+
The Galois differential groups of some typical differential algebraic extensions have the following form.
  
3) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960171.png" /> be elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960172.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960173.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960174.png" /> is said to be Weierstrass over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960175.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960176.png" /> satisfies the system of equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960177.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960178.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960179.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960180.png" />. The extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960181.png" /> is strongly normal over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960182.png" />, but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960183.png" /> is transcendental over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960184.png" />, it is not a Picard–Vessiot extension. There is a monomorphism
+
1) Let $  {\mathcal G} = F \langle  \alpha \rangle $,
 +
where  $  \alpha $
 +
satisfies the system of equations $  \delta _{i} \alpha = a _{i} \alpha $,
 +
$  \delta _{i} \in \Delta $,  
 +
$  a _{i} \in F $,  
 +
$  i = 1 \dots m $,  
 +
and let the fields of constants of  $  {\mathcal G} $
 +
and  $  F $
 +
coincide. Then  $  {\mathcal G} $
 +
is a Picard–Vessiot extension of  $  F $
 +
and the Galois differential group  $  \mathop{\rm Gal}\nolimits ( {\mathcal G} /F \  ) $
 +
is a subgroup of the multiplicative group of  $  K $(
 +
that is,  $  \mathop{\rm GL}\nolimits (1,\  K) = K ^{*} $).  
 +
If  $  \alpha $
 +
is algebraic, it satisfies an equation of the form  $  y ^{d} - b = 0 $,
 +
where  $  b \in F $
 +
and  $  \mathop{\rm Gal}\nolimits ( {\mathcal G} /F \  ) = \mathbf Z _{d} $(
 +
the group of  $  d $-
 +
th roots of unity). In this case, $  {\mathcal G} $
 +
is called an extension of  $  F $
 +
by an exponent.
  
<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/e/e036/e036960/e036960185.png" /></td> </tr></table>
+
2) Let  $  {\mathcal G} = F \langle  \alpha \rangle $,
 +
where  $  \alpha $
 +
satisfies the system of equations  $  \delta _{i} \alpha = a _{i} $,
 +
$  \delta _{i} \in \Delta $,
 +
$  a _{i} \in F $,
 +
$  i = 1 \dots m $(
 +
such an element  $  \alpha $
 +
is called primitive over  $  F \  $),
 +
and let the field of constants of  $  F \langle  \alpha \rangle $
 +
coincide with  $  C $.  
 +
If  $  \alpha \notin F $,
 +
then  $  \alpha $
 +
is transcendental over  $  F $.  
 +
The resulting extension is a Picard–Vessiot extension, and the Galois group  $  \mathop{\rm Gal}\nolimits (F \langle  \alpha \rangle /F \  ) $
 +
is isomorphic to the additive group of  $  K $.
 +
Such extensions are called extensions by an integral.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960186.png" /> is the group of points on the cubic curve
+
3) Let  $  g _{2} ,\  g _{3} $
 +
be elements of  $  C $
 +
such that  $  g _{2} ^{3} - 27g _{3} ^{2} \neq 0 $.  
 +
An element  $  \alpha \in U $
 +
is said to be Weierstrass over  $  F $
 +
if  $  \alpha $
 +
satisfies the system of equations  $  ( \delta _{i} \alpha ) ^{2} - a _{i} ^{2} (4 \alpha ^{3} - g _{2} \alpha - g _{3} ) $,
 +
$  \delta _{i} \in \Delta $,
 +
$  a _{i} \in F $,
 +
$  1 \leq i \leq m $.  
 +
The extension  $  {\mathcal G} = F \langle  \alpha \rangle $
 +
is strongly normal over  $  F $,
 +
but if  $  \alpha $
 +
is transcendental over  $  F $,
 +
it is not a Picard–Vessiot extension. There is a monomorphism $$
 +
c: \  \mathop{\rm Gal}\nolimits (F \langle  \alpha \rangle /F \  )  \rightarrow  W _{K} ,
 +
$$
 +
where  $  W _{K} $
 +
is the group of points on the cubic curve $$
 +
X _{0} X _{2} ^{2} -
 +
(4X _{1} ^{3} -
 +
g _{2} X _{0} ^{2} X _{1} -
 +
g _{3} X _{0} ^{3} )  =  0.
 +
$$
 +
If  $  \alpha $
 +
is transcendental over  $  F $,
 +
then  $  c $
 +
is an isomorphism.
  
<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/e/e036/e036960/e036960187.png" /></td> </tr></table>
+
4) Let  $  F $
 +
be a differential field,  $  a _{1} \dots a _{n} \in F $,
 +
and let  $  ( \eta _{1} \dots \eta _{n} ) $
 +
be the fundamental set of zeros of the equation  $  y ^{(n)} + a _{1} y ^ {(n - 1)} + \dots + a _{n} y = 0 $,
 +
which generates the Picard–Vessiot extension of  $  F $.
 +
The Galois group  $  \mathop{\rm Gal}\nolimits (F \langle  \eta _{1} \dots \eta _{n} \rangle /F \  ) $
 +
is contained in  $  \mathop{\rm SL}\nolimits (n,\  K) $
 +
if and only if the equation  $  y ^ \prime  + a _{1} y = 0 $
 +
has a non-trivial zero in  $  F $.
 +
In particular, if  $  F = \mathbf C (x) $
 +
is the differential field of rational functions of one complex variable with differentiation  $  d/dx $
 +
and  $  B _ \nu  = y ^{\prime\prime} + x ^{-1} + (1 - \nu ^{2} x ^{-2} ) y $
 +
is the Bessel differential polynomial, then the Galois group of the corresponding extension coincides with  $  \mathop{\rm SL}\nolimits (2,\  K) $
 +
for  $  \nu - 1/2 \notin \mathbf Z $.  
 +
If  $  \nu - 1/2 \in \mathbf Z $,
 +
then the Galois group coincides with  $  K ^{*} $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960188.png" /> is transcendental over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960189.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960190.png" /> is an isomorphism.
 
  
4) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960191.png" /> be a differential field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960192.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960193.png" /> be the fundamental set of zeros of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960194.png" />, which generates the Picard–Vessiot extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960195.png" />. The Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960196.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960197.png" /> if and only if the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960198.png" /> has a non-trivial zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960199.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960200.png" /> is the differential field of rational functions of one complex variable with differentiation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960202.png" /> is the Bessel differential polynomial, then the Galois group of the corresponding extension coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960203.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960204.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960205.png" />, then the Galois group coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960206.png" />.
+
For all positive integers  $  n $
 +
one can construct extensions of differential fields  $  {\mathcal G} \supset F $
 +
such that  $  \mathop{\rm Gal}\nolimits ( {\mathcal G} /F \  ) \approx  \mathop{\rm GL}\nolimits (n,\  K) $.
  
For all positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960207.png" /> one can construct extensions of differential fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960208.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960209.png" />.
 
  
 
A [[Galois correspondence|Galois correspondence]] exists between the set of differential subfields of a strongly normal extension and the set of algebraic subgroups of its Galois group.
 
A [[Galois correspondence|Galois correspondence]] exists between the set of differential subfields of a strongly normal extension and the set of algebraic subgroups of its Galois group.
Line 45: Line 257:
 
As in ordinary Galois theory, two general problems are of interest in the differential case.
 
As in ordinary Galois theory, two general problems are of interest in the differential case.
  
a) The direct problem: Given a strongly normal extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960210.png" /> of a differential field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960211.png" />, determine its Galois group.
+
a) The direct problem: Given a strongly normal extension $  {\mathcal G} $
 +
of a differential field $  F $,  
 +
determine its Galois group.
  
b) The converse problem: Given a differential field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960212.png" /> and an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960213.png" />, describe the set of strongly normal extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960214.png" /> with Galois group isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e036960215.png" /> (in particular, determine if it is non-empty).
+
b) The converse problem: Given a differential field $  F $
 +
and an algebraic group $  G $,  
 +
describe the set of strongly normal extensions of $  F $
 +
with Galois group isomorphic to $  G $(
 +
in particular, determine if it is non-empty).
  
 
There is another way of generalizing normality in the case of extensions of differential fields and of constructing a differential Galois theory; this uses methods of differential geometry [[#References|[4]]].
 
There is another way of generalizing normality in the case of extensions of differential fields and of constructing a differential Galois theory; this uses methods of differential geometry [[#References|[4]]].

Latest revision as of 21:02, 17 December 2019


$ F _{0} $


A differential field $ F \supset F _{0} $ with a set $ \Delta $ of differentiations such that the set of restrictions of the elements of $ \Delta $ to $ F _{0} $ coincides with the set of differentiations on $ F _{0} $. In turn, $ F _{0} $ is a differential subfield of $ F $.


The intersection of any set of differential subfields of $ F $ is again a differential subfield of $ F $. For any set of elements $ \Sigma \subset F $ there is a smallest differential subfield of $ F $ containing all the elements of $ \Sigma $ and $ F _{0} $; it is denoted by $ F _{0} \langle \Sigma \rangle $ and is called the extension of the field $ F _{0} $ generated by the set $ \Sigma $( and $ \Sigma $ is called a set, or family, of generators of the extension $ F _{0} \langle \Sigma \rangle $ over $ F _{0} $). The extension is said to be finitely generated if it has a finite set of generators, and is called simply generated if the set of generators consists of one element. If $ F _{1} $ and $ F _{2} $ are two differential subfields of $ F $, then the subfield $$ F _{1} F _{2} = F _{1} \langle F _{2} \rangle = F _{1} (F _{2} ) = F _{2} (F _{1} ) = F _{2} \langle F _{1} \rangle, $$ is a differential subfield of $ F $, called the join of the fields $ F _{1} $ and $ F _{2} $.


Let $ \Theta $ be the free commutative semi-group generated by $ \Delta $( its elements are called differential operators). A family $ ( \alpha _{i} ) _ {i \in I} $ of elements of $ F $ is said to be differentially algebraically dependent over $ F _{0} \subset F $ if the family $ ( \theta \alpha _{i} ) _ {i \in I,\ \theta \in \Theta} $ is algebraically dependent over $ F _{0} $. In the opposite case, the family $ ( \alpha _{i} ) _ {i \in I} $ is called differentially algebraically independent over $ F _{0} $, or a family of differential indeterminates over $ F _{0} $. One says that the elements of $ ( \alpha _{i} ) _ {i \in I} $ are differentially separably dependent over $ F _{0} $ if the family $ ( \theta \alpha _{i} ) _ {i \in I,\ \theta \in \Theta} $ is separably dependent over $ F _{0} $. In the opposite case the family $ ( \alpha _{i} ) _ {i \in I} $ is called differentially separably independent over $ F _{0} $.


An extension $ F $ is called differentially algebraic over $ F _{0} $ if every element of $ F $ is differentially algebraic over $ F _{0} $. Similarly, $ F $ is called differentially separable over $ F _{0} $ if every element of $ F $ is differentially separable over $ F _{0} $. The theorem on the primitive element applies to differential extensions: If the set $ \Theta $ is independent over $ F _{0} $, then every finitely-generated differentially-separable extension $ F $ of $ F _{0} $ is generated by one element.

Let $ J $ be a given set and let $ F _{0} [(y _ {j \theta} ) _ {j \in J,\ \theta \in \Theta} ] $ be the polynomial algebra over $ F _{0} $ in the family of indeterminates $ (y _ {j \theta} ) _ {j \in J,\ \theta \in \Theta} $, with index set $ J \times \Theta $. Any differentiation $ \delta \in \Delta $ of $ F _{0} $ extends in a unique way to a differentiation of $ F _{0} [(y _ {j \theta} ) _ {j \in J,\ \theta \in \Theta} ] $ sending $ y _ {j \theta} $ to $ y _ {j \delta \theta} $. This differential ring is called the ring of differential polynomials in the differential indeterminates $ y _{j} $, $ j \in J $, and is denoted by $ F _{0} \{ (y _{j} ) _ {j \in J} \} $. Its differential field of fractions (i.e. the field of fractions with extended differentiations) is denoted by $ F _{0} \langle (y _{j} ) _ {j \in J} \rangle $, and the elements of this field are called differential functions over $ F _{0} $ in the differential indeterminates $ (y _{j} ) _ {j \in J} $. For ordinary differential fields an analogue of the Lüroth theorem holds: If $ F $ is an arbitrary differential extension of a differential field $ F _{0} $ contained in $ F _{0} \langle u \rangle $, then $ F $ contains an element $ v $ such that $ F = F _{0} \langle v \rangle $.


For any differential field $ F $ there is a separable semi-universal extension, i.e. an extension containing every finitely-generated separable extension of $ F $. Moreover, there exists a separable universal extension $ U $, i.e. an extension which is semi-universal over every finitely-generated extension of $ F $ contained in $ U $.


In the theory of differential fields there is no direct analogue of the notion of an algebraically closed field in ordinary field theory. To a certain extent, their role is played by constrainedly closed fields. The main property of such a field $ F $ is that any finite system of algebraic differential equations and inequalities with coefficients in $ F $ having a solution that is rational over some field extension of $ F $ has a solution that is rational over $ F $. A family $ \eta = (n _{j} ) _ {j \in J} $ of elements of some extension of $ F $ is called constrained over $ F $ if there is a differential polynomial $ c \in F \{ (y _{j} ) _ {j \in J} \} $ such that $ c ( \eta ) \neq 0 $ and $ c ( \eta ^ \prime ) = 0 $ for any non-generic differential specialization $ \eta ^ \prime $ of the point $ \eta $ over $ F $. An extension $ {\mathcal G} $ of $ F $ is called constrained over $ F $ if any finite set of elements $ \eta _{1} \dots \eta _{n} \in {\mathcal G} $ is constrained over $ F $. This is equivalent to saying that an arbitrary element of $ {\mathcal G} $ is constrained over $ F $. A differential field having no non-trivial constrained extensions is called constrainedly closed. An example of such a field is the universal differential field of characteristic zero (the universal field extension of the field of rational numbers $ \mathbf Q $). Any differential field $ F $ of characteristic zero has a constrained closure, i.e. a constrainedly closed extension of $ F $ which is contained in any other constrainedly closed extension of $ F $.


The notion of a normal extension in ordinary field theory carries over to differential algebra in various ways. In differential Galois theory, a fundamental role is played by strongly normal extensions. Let $ U $ be the fixed universal differential field of characteristic 0 with field of constants $ K $. All the differential fields encountered below are assumed to lie in $ U $ and all isomorphisms are assumed to be differential isomorphisms, that is, they commute with the operators in $ \Delta $. Let $ F $ and $ {\mathcal G} $ be differential fields over which $ U $ is universal. Let $ C $ be the field of constants of $ {\mathcal G} $. An isomorphism $ \sigma $ leaves invariant each element of $ C $, $ \sigma {\mathcal G} \subset {\mathcal G} K $, and $ {\mathcal G} \subset \sigma {\mathcal G} K $( that is, $ {\mathcal G} K = {\mathcal G} \sigma K $). A strongly normal extension of $ F $ is a finitely-generated extension $ {\mathcal G} $ of $ F $ such that every isomorphism of $ {\mathcal G} $ over $ F $ is strong. Strongly normal extensions are constrained. The set of strong isomorphisms of a strongly normal extension $ {\mathcal G} $ over $ F $ has the natural structure of an algebraic group, defined over $ K $( and denoted by $ \mathop{\rm Gal}\nolimits ( {\mathcal G} /F \ ) $). This is the Galois differential group of the extension $ {\mathcal G} /F $. A special case of strongly normal extensions is given by the Picard–Vessiot extensions, i.e. extensions that preserve the field of constants and result from the adjunction to $ F $ of a basis for the solutions of some system of homogeneous linear differential equations with coefficients in $ F $. For extensions of this type, $ \mathop{\rm Gal}\nolimits ( {\mathcal G} /F \ ) $ is an algebraic matrix group, i.e. an algebraic subgroup of the group $ \mathop{\rm GL}\nolimits (n,\ K) $ for some integer $ n > 0 $.


The Galois differential groups of some typical differential algebraic extensions have the following form.

1) Let $ {\mathcal G} = F \langle \alpha \rangle $, where $ \alpha $ satisfies the system of equations $ \delta _{i} \alpha = a _{i} \alpha $, $ \delta _{i} \in \Delta $, $ a _{i} \in F $, $ i = 1 \dots m $, and let the fields of constants of $ {\mathcal G} $ and $ F $ coincide. Then $ {\mathcal G} $ is a Picard–Vessiot extension of $ F $ and the Galois differential group $ \mathop{\rm Gal}\nolimits ( {\mathcal G} /F \ ) $ is a subgroup of the multiplicative group of $ K $( that is, $ \mathop{\rm GL}\nolimits (1,\ K) = K ^{*} $). If $ \alpha $ is algebraic, it satisfies an equation of the form $ y ^{d} - b = 0 $, where $ b \in F $ and $ \mathop{\rm Gal}\nolimits ( {\mathcal G} /F \ ) = \mathbf Z _{d} $( the group of $ d $- th roots of unity). In this case, $ {\mathcal G} $ is called an extension of $ F $ by an exponent.

2) Let $ {\mathcal G} = F \langle \alpha \rangle $, where $ \alpha $ satisfies the system of equations $ \delta _{i} \alpha = a _{i} $, $ \delta _{i} \in \Delta $, $ a _{i} \in F $, $ i = 1 \dots m $( such an element $ \alpha $ is called primitive over $ F \ $), and let the field of constants of $ F \langle \alpha \rangle $ coincide with $ C $. If $ \alpha \notin F $, then $ \alpha $ is transcendental over $ F $. The resulting extension is a Picard–Vessiot extension, and the Galois group $ \mathop{\rm Gal}\nolimits (F \langle \alpha \rangle /F \ ) $ is isomorphic to the additive group of $ K $. Such extensions are called extensions by an integral.

3) Let $ g _{2} ,\ g _{3} $ be elements of $ C $ such that $ g _{2} ^{3} - 27g _{3} ^{2} \neq 0 $. An element $ \alpha \in U $ is said to be Weierstrass over $ F $ if $ \alpha $ satisfies the system of equations $ ( \delta _{i} \alpha ) ^{2} - a _{i} ^{2} (4 \alpha ^{3} - g _{2} \alpha - g _{3} ) $, $ \delta _{i} \in \Delta $, $ a _{i} \in F $, $ 1 \leq i \leq m $. The extension $ {\mathcal G} = F \langle \alpha \rangle $ is strongly normal over $ F $, but if $ \alpha $ is transcendental over $ F $, it is not a Picard–Vessiot extension. There is a monomorphism $$ c: \ \mathop{\rm Gal}\nolimits (F \langle \alpha \rangle /F \ ) \rightarrow W _{K} , $$ where $ W _{K} $ is the group of points on the cubic curve $$ X _{0} X _{2} ^{2} - (4X _{1} ^{3} - g _{2} X _{0} ^{2} X _{1} - g _{3} X _{0} ^{3} ) = 0. $$ If $ \alpha $ is transcendental over $ F $, then $ c $ is an isomorphism.

4) Let $ F $ be a differential field, $ a _{1} \dots a _{n} \in F $, and let $ ( \eta _{1} \dots \eta _{n} ) $ be the fundamental set of zeros of the equation $ y ^{(n)} + a _{1} y ^ {(n - 1)} + \dots + a _{n} y = 0 $, which generates the Picard–Vessiot extension of $ F $. The Galois group $ \mathop{\rm Gal}\nolimits (F \langle \eta _{1} \dots \eta _{n} \rangle /F \ ) $ is contained in $ \mathop{\rm SL}\nolimits (n,\ K) $ if and only if the equation $ y ^ \prime + a _{1} y = 0 $ has a non-trivial zero in $ F $. In particular, if $ F = \mathbf C (x) $ is the differential field of rational functions of one complex variable with differentiation $ d/dx $ and $ B _ \nu = y ^{\prime\prime} + x ^{-1} + (1 - \nu ^{2} x ^{-2} ) y $ is the Bessel differential polynomial, then the Galois group of the corresponding extension coincides with $ \mathop{\rm SL}\nolimits (2,\ K) $ for $ \nu - 1/2 \notin \mathbf Z $. If $ \nu - 1/2 \in \mathbf Z $, then the Galois group coincides with $ K ^{*} $.


For all positive integers $ n $ one can construct extensions of differential fields $ {\mathcal G} \supset F $ such that $ \mathop{\rm Gal}\nolimits ( {\mathcal G} /F \ ) \approx \mathop{\rm GL}\nolimits (n,\ K) $.


A Galois correspondence exists between the set of differential subfields of a strongly normal extension and the set of algebraic subgroups of its Galois group.

As in ordinary Galois theory, two general problems are of interest in the differential case.

a) The direct problem: Given a strongly normal extension $ {\mathcal G} $ of a differential field $ F $, determine its Galois group.

b) The converse problem: Given a differential field $ F $ and an algebraic group $ G $, describe the set of strongly normal extensions of $ F $ with Galois group isomorphic to $ G $( in particular, determine if it is non-empty).

There is another way of generalizing normality in the case of extensions of differential fields and of constructing a differential Galois theory; this uses methods of differential geometry [4].

References

[1] J.F. Ritt, "Differential algebra" , Amer. Math. Soc. (1950) MR0035763 Zbl 0037.18402
[2] E.R. Kolchin, "Differential algebra and algebraic groups" , Acad. Press (1973) MR0568864 Zbl 0264.12102
[3] I. Kaplansky, "An introduction to differential algebra" , Hermann (1976) MR0460303 Zbl 0954.12500 Zbl 0089.02301 Zbl 0083.03301
[4] J.F. Pommaret, "Differential Galois theory" , Gordon & Breach (1983) MR0720863 MR0712173 Zbl 0539.12013 Zbl 0528.12019
How to Cite This Entry:
Extension of a differential field. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Extension_of_a_differential_field&oldid=44301
This article was adapted from an original article by A.V. MikhalevE.V. Pankrat'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article