Extension of a differential field
A differential field with a set of differentiations such that the set of restrictions of the elements of to coincides with the set of differentiations on . In turn, is a differential subfield of .
The intersection of any set of differential subfields of is again a differential subfield of . For any set of elements there is a smallest differential subfield of containing all the elements of and ; it is denoted by and is called the extension of the field generated by the set (and is called a set, or family, of generators of the extension over ). 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 and are two differential subfields of , then the subfield
is a differential subfield of , called the join of the fields and .
Let be the free commutative semi-group generated by (its elements are called differential operators). A family of elements of is said to be differentially algebraically dependent over if the family is algebraically dependent over . In the opposite case, the family is called differentially algebraically independent over , or a family of differential indeterminates over . One says that the elements of are differentially separably dependent over if the family is separably dependent over . In the opposite case the family is called differentially separably independent over .
An extension is called differentially algebraic over if every element of is differentially algebraic over . Similarly, is called differentially separable over if every element of is differentially separable over . The theorem on the primitive element applies to differential extensions: If the set is independent over , then every finitely-generated differentially-separable extension of is generated by one element.
Let be a given set and let be the polynomial algebra over in the family of indeterminates , with index set . Any differentiation of extends in a unique way to a differentiation of sending to . This differential ring is called the ring of differential polynomials in the differential indeterminates , , and is denoted by . Its differential field of fractions (i.e. the field of fractions with extended differentiations) is denoted by , and the elements of this field are called differential functions over in the differential indeterminates . For ordinary differential fields an analogue of the Lüroth theorem holds: If is an arbitrary differential extension of a differential field contained in , then contains an element such that .
For any differential field there is a separable semi-universal extension, i.e. an extension containing every finitely-generated separable extension of . Moreover, there exists a separable universal extension , i.e. an extension which is semi-universal over every finitely-generated extension of contained in .
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 is that any finite system of algebraic differential equations and inequalities with coefficients in having a solution that is rational over some field extension of has a solution that is rational over . A family of elements of some extension of is called constrained over if there is a differential polynomial such that and for any non-generic differential specialization of the point over . An extension of is called constrained over if any finite set of elements is constrained over . This is equivalent to saying that an arbitrary element of is constrained over . 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 ). Any differential field of characteristic zero has a constrained closure, i.e. a constrainedly closed extension of which is contained in any other constrainedly closed extension of .
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 be the fixed universal differential field of characteristic 0 with field of constants . All the differential fields encountered below are assumed to lie in and all isomorphisms are assumed to be differential isomorphisms, that is, they commute with the operators in . Let and be differential fields over which is universal. Let be the field of constants of . An isomorphism leaves invariant each element of , , and (that is, ). A strongly normal extension of is a finitely-generated extension of such that every isomorphism of over is strong. Strongly normal extensions are constrained. The set of strong isomorphisms of a strongly normal extension over has the natural structure of an algebraic group, defined over (and denoted by ). This is the Galois differential group of the extension . 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 of a basis for the solutions of some system of homogeneous linear differential equations with coefficients in . For extensions of this type, is an algebraic matrix group, i.e. an algebraic subgroup of the group for some integer .
The Galois differential groups of some typical differential algebraic extensions have the following form.
1) Let , where satisfies the system of equations , , , , and let the fields of constants of and coincide. Then is a Picard–Vessiot extension of and the Galois differential group is a subgroup of the multiplicative group of (that is, ). If is algebraic, it satisfies an equation of the form , where and (the group of -th roots of unity). In this case, is called an extension of by an exponent.
2) Let , where satisfies the system of equations , , , (such an element is called primitive over ), and let the field of constants of coincide with . If , then is transcendental over . The resulting extension is a Picard–Vessiot extension, and the Galois group is isomorphic to the additive group of . Such extensions are called extensions by an integral.
3) Let be elements of such that . An element is said to be Weierstrass over if satisfies the system of equations , , , . The extension is strongly normal over , but if is transcendental over , it is not a Picard–Vessiot extension. There is a monomorphism
where is the group of points on the cubic curve
If is transcendental over , then is an isomorphism.
4) Let be a differential field, , and let be the fundamental set of zeros of the equation , which generates the Picard–Vessiot extension of . The Galois group is contained in if and only if the equation has a non-trivial zero in . In particular, if is the differential field of rational functions of one complex variable with differentiation and is the Bessel differential polynomial, then the Galois group of the corresponding extension coincides with for . If , then the Galois group coincides with .
For all positive integers one can construct extensions of differential fields such that .
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 of a differential field , determine its Galois group.
b) The converse problem: Given a differential field and an algebraic group , describe the set of strongly normal extensions of with Galois group isomorphic to (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 .
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