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**The differential Galois theory of strongly normal extensions.**
*(English)*
Zbl 1036.12005

Let \(E \supseteq F\) be an extension of \(\Delta\)-fields, where \(\Delta=\{\delta_1, \dots, \delta_n\}\) is a set of commuting derivations. Assume that the characteristic is zero, that \(E\) is finitely generated over \(F\) as a differential field, and that the field \(C\) of constants of \(F\) is algebraically closed. The author uses the \(\Delta\)-ring \(R=E \otimes_F E\) and its differential spectrum \(X\) to reconstruct E. Kolchin’s Galois theory of strongly normal extensions in a self-contained modern context.

A \(\Delta\)-isomorphism of \(E\) over \(F\) is a \(\Delta\)-homomorphism \(\sigma:E \to G\) to some \(\Delta\)-field extension \(G \supseteq F\) such that \(\sigma\) is the identity on \(F\). There is an induced map \(R \to G\) by \(a \otimes b \mapsto a\sigma(b)\) whose kernel \(P_\sigma\) is a prime \(\Delta\)-ideal and hence belongs to \(X\).

Let \(C(\sigma)\) denote the constants of \(E\sigma(E)\). Call \(\sigma\) strong if \(\sigma\) restricted to the constants of \(E\) is the identity and if \(E\sigma(E)=EC(\sigma)=\sigma(E)C(\sigma)\). Call \(P \in X\) strong if \(c \otimes 1 -1 \otimes c \in P\) for all constants \(c\) of \(E\), and if the quotient field \(\kappa(P)\) of \(R/P\), which is the compositum of the images of \(E\) under the maps \(a \mapsto a \otimes 1\) and \(a \mapsto 1 \otimes a\), is the compositum of either of these images with the field of constants \(C(P)\) of \(\kappa(P)\). Then \(\sigma\) is strong if and only if \(P_\sigma\) is strong. The extension \(E \supseteq F\) is then called strongly normal if every \(\Delta\)-homomorphism of \(E\) over \(F\) is strong.

Now assume that \(E \supseteq F\) is strongly normal and let \(\text{Gal}(E/F)\) denote the group of \(\Delta\)-automorphisms of \(E\) over \(F\). The elements of \(\text{Gal}(E/F)\) are \(\Delta\)-homomorphisms of \(E\) over \(F\) and hence correspond to elements of \(X\); the author shows that the corresponding prime ideals are precisely the maximal \(\Delta\)-ideals of \(R\). Thus \(\text{Gal}(E/F)\) can be identified with the set of maximal ideals of \(X\).

He introduces the notion of a differential coring and shows that \(R \to R \otimes_E R\) by \(a \otimes b \mapsto (a \otimes 1) \otimes (1 \otimes a)\) makes \(R\) a \(\Delta\)-coring. If \(K\), \(E \supseteq K \supseteq F\), is an intermediate \(\Delta\)-field, then the kernel \(C(K)\) of \(E \otimes_F E \to E \otimes_K E\) is a \(\Delta\)-coideal, and if \(A \supset R\) is a \(\Delta\)-coideal then \(I(A)=\{x \in E \mid a \otimes 1 - 1 \otimes a \in A \}\) is an intermediate \(\Delta\)-field, and the correspondences \(K \to C(K)\) and \(A \to I(A)\) are inverse bijections. The Kolchin topology on \(X\) induces a topology on \(\text{Gal}(E/F)\), and the author shows, using the above bijections, that the closed subgroups are precisely those of the form \(\text{Gal}(E/K)\) for intermediate fields \(K\), thus obtaining the (first) Fundamental Theorem of Galois Theory. (He obtains the second fundamental theorem as well.)

To complete the Galois theory of strongly normal extensions, it remains to see that \(\text{Gal}(E/F)\) has the structure of an algebraic group over the field \(C\) of constants. This the author accomplishes by showing that the \(\Delta\)-scheme \(X\) is such that the associated \(C\) scheme \(X^{\Delta}\) is a group scheme over \(C\) whose group of \(C\) points is bijective with \(\text{Gal}(E/F)\), the bijection preserving the topology induced from the Kolchin topology and hence consistent with the fundamental theorem.

A \(\Delta\)-isomorphism of \(E\) over \(F\) is a \(\Delta\)-homomorphism \(\sigma:E \to G\) to some \(\Delta\)-field extension \(G \supseteq F\) such that \(\sigma\) is the identity on \(F\). There is an induced map \(R \to G\) by \(a \otimes b \mapsto a\sigma(b)\) whose kernel \(P_\sigma\) is a prime \(\Delta\)-ideal and hence belongs to \(X\).

Let \(C(\sigma)\) denote the constants of \(E\sigma(E)\). Call \(\sigma\) strong if \(\sigma\) restricted to the constants of \(E\) is the identity and if \(E\sigma(E)=EC(\sigma)=\sigma(E)C(\sigma)\). Call \(P \in X\) strong if \(c \otimes 1 -1 \otimes c \in P\) for all constants \(c\) of \(E\), and if the quotient field \(\kappa(P)\) of \(R/P\), which is the compositum of the images of \(E\) under the maps \(a \mapsto a \otimes 1\) and \(a \mapsto 1 \otimes a\), is the compositum of either of these images with the field of constants \(C(P)\) of \(\kappa(P)\). Then \(\sigma\) is strong if and only if \(P_\sigma\) is strong. The extension \(E \supseteq F\) is then called strongly normal if every \(\Delta\)-homomorphism of \(E\) over \(F\) is strong.

Now assume that \(E \supseteq F\) is strongly normal and let \(\text{Gal}(E/F)\) denote the group of \(\Delta\)-automorphisms of \(E\) over \(F\). The elements of \(\text{Gal}(E/F)\) are \(\Delta\)-homomorphisms of \(E\) over \(F\) and hence correspond to elements of \(X\); the author shows that the corresponding prime ideals are precisely the maximal \(\Delta\)-ideals of \(R\). Thus \(\text{Gal}(E/F)\) can be identified with the set of maximal ideals of \(X\).

He introduces the notion of a differential coring and shows that \(R \to R \otimes_E R\) by \(a \otimes b \mapsto (a \otimes 1) \otimes (1 \otimes a)\) makes \(R\) a \(\Delta\)-coring. If \(K\), \(E \supseteq K \supseteq F\), is an intermediate \(\Delta\)-field, then the kernel \(C(K)\) of \(E \otimes_F E \to E \otimes_K E\) is a \(\Delta\)-coideal, and if \(A \supset R\) is a \(\Delta\)-coideal then \(I(A)=\{x \in E \mid a \otimes 1 - 1 \otimes a \in A \}\) is an intermediate \(\Delta\)-field, and the correspondences \(K \to C(K)\) and \(A \to I(A)\) are inverse bijections. The Kolchin topology on \(X\) induces a topology on \(\text{Gal}(E/F)\), and the author shows, using the above bijections, that the closed subgroups are precisely those of the form \(\text{Gal}(E/K)\) for intermediate fields \(K\), thus obtaining the (first) Fundamental Theorem of Galois Theory. (He obtains the second fundamental theorem as well.)

To complete the Galois theory of strongly normal extensions, it remains to see that \(\text{Gal}(E/F)\) has the structure of an algebraic group over the field \(C\) of constants. This the author accomplishes by showing that the \(\Delta\)-scheme \(X\) is such that the associated \(C\) scheme \(X^{\Delta}\) is a group scheme over \(C\) whose group of \(C\) points is bijective with \(\text{Gal}(E/F)\), the bijection preserving the topology induced from the Kolchin topology and hence consistent with the fundamental theorem.

Reviewer: Andy R. Magid (Norman)

### Keywords:

differential Galois theory
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\textit{J. J. Kovacic}, Trans. Am. Math. Soc. 355, No. 11, 4475--4522 (2003; Zbl 1036.12005)

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