Galois Correspondence Theorem for Picard-Vessiot Extensions

For a homogeneous linear differential equation $\mathcal{L}(Y)=0$ defined over a differential field $K$ with field of constants $C$, a Picard-Vessiot extension is a differential field $L$, differentially generated over $K$ by a fundamental system of solutions of $\mathcal{L}(Y)=0$ and with constant field equal to $C$. A classical result states that the Picard-Vessiot extension exists and is unique up to $K$-differential isomorphism in the case $C$ algebraically closed (see [Kolchin1948], [Crespo and Hajto2011] or [Put and Singer2003]). Recently, an existence and uniqueness result for Picard-Vessiot extensions has been established in the case when the differential field $K$ is a formally real field (resp. a formally $p$-adic field) with real closed (resp. $p$-adically closed of the same rank than $K$) field of constants $C$ (see [Crespo et al.2015]). In [Crespo et al.2013] we presented a Galois correspondence theorem for Picard-Vessiot extensions of formally real differential fields with real closed field of constants. In this paper we establish a Galois correspondence theorem for general Picard-Vessiot extensions, i.e. without assuming the field of constants of the base field to be algebraically closed, valid in particular for Picard-Vessiot extensions of formally $p$-adic differential fields with $p$-adically closed field of constants.

Kolchin introduced the concept of strongly normal differential field extension and obtained a satisfactory Galois correspondence theorem for this class of extensions without assuming the field of constants of the differential base field to be algebraically closed (see [Kolchin1973], Chapter VI). Note that, for a strongly normal extension $L|K$, in the case when the constant field of $K$ is not algebraically closed, the differential Galois group is no longer the group $\mathrm{DAut}_{K}L$ of $K$-differential automorphisms of $L$, rather one has to consider as well $K$-differential morphisms of $L$ in larger differential fields. It is worth noting that a Picard-Vessiot extension is always strongly normal. The following simple example illustrates the necessity to change the definition of the differential Galois group when the field of constants is not algebraically closed. Let $q$ be an odd prime number and $C$ a field not containing the $q$-th roots of unity (e.g. $C=\mathbb{Q}_{p}$, with $q\nmid p-1$). Let $k=C(t)$ be the field of rational functions over $C$ in the variable $t$ endowed with the derivation determined by $t^{\prime}=1$. Let $L=k(y)$ be the field of rational functions over $k$ in the variable $y$ and extend the derivation to $L$ by $y^{\prime}=y$. Let $K=k(y^{q})$. Then $L|K$ is clearly a Picard-Vessiot extension for the differential equation $Y^{\prime}=Y$ with $[L:K]=q$ and $\mathrm{DAut}_{K}L=\mathrm{Aut}_{K}L=\{Id\}$. In the case of Picard-Vessiot extensions, we can adopt a definition of the differential Galois group inspired by Kolchin’s but simpler than his one. We obtain then a Galois correspondence theorem which classifies intermediate differential fields of a Picard-Vessiot extension in terms of its differential Galois group. The fact that Kolchin used the notion of differential universal extension and Weil’s algebraic geometry, which has been later replaced by Grothendieck’s one, has led several authors to redo Kolchin’s theory using the more modern language of schemes or Hopf algebras. However, as the knowledge of scheme and Hopf algebra theories is not extended to non-algebraists, we have chosen a simpler approach which will make the theory accessible to a wider range of mathematicians.

In this paper, we shall deal with fields of characteristic 0. For the sake of simplicity in the exposition we consider ordinary differential fields.

We recall now the precise definition of Picard-Vessiot extension.

As mentioned in the introduction, a Picard-Vessiot extension is strongly normal. Hence, the fundamental theorem established by [Kolchin1973], chapter VI applies to Picard-Vessiot extensions. However, for a strongly normal extension $L|K$, Kolchin defines the differential Galois group $\mathrm{DGal}(L|K)$ by means of differential $K$-isomorphisms of $L$ in the differential universal extension $U$ of $L$. He obtains then that $\mathrm{DGal}(L|K)$ has the structure of an algebraic group defined over the field of constants $C_{U}$ of $U$. Afterwards, by using the notion of specialization in Weil’s algebraic geometry, he proves that there exists an algebraic group $G$ defined over the field of constants $C$ of $K$ such that $G(C_{U})=\mathrm{DGal}(L|K)$. It is worth noting that Kovacic in [Kovacic2003]; [Kovacic2006] established a theory of strongly normal extensions which avoids the differential universal extension by using tensor products. He defines the differential Galois group using a differential scheme. [Umemura1996] introduced the notion of automorphic differential field extensions, which allows a finite extension of the constant field, and includes both strongly normal extensions and almost classically Galois extensions (see [Umemura1996] Corollary (2.26)), a special class of Hopf-Galois finite extensions introduced in [Greither and Pareigis1987]. Here a finite extension of fields is considered as a differential field extension using the trivial derivation. The differential Galois group of an automorphic differential field extension is defined as a group functor. Using Kovacic’s theory of strongly normal extensions, [Wibmer2014] gave an intrinsic characterization of a Picard-Vessiot extension $L|K$ in terms of $K$-differential embeddings of $L$ in differential $L$-algebras.

In this section, we give a more natural definition of the differential Galois group of a Picard-Vessiot extension, which avoids the use both of the differential universal extension and of scheme theory. We endow the differential Galois group with a linear algebraic group structure over the field of constants $C$ and establish a Galois correspondence theorem in our setting.