Extension of a differential field

$ 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