Summary
The partial differential fields most suited for the purpose of construction of Galois theory for the family (1) are endowed with the symmetric bilinear form (2iv) and are called α-differential fields. In Section 1 are defined certain algebraic notions related to the symmetric bilinear form (2iv) and which are necessary for the construction of any Galois theory. Necessary and sufficient condition for the extension of the domain of the operator Δα (this operator is not a derivation although it commutes with the partial derivations of the α-differential field) from an α-differential fieldK to a finitely generated α-differential extension field is given in Theorem 1.
Section 2 defines the notion of α-differential mapping as linear mappings which preserve the symmetric bilinear form and commute with the partial derivations. The group properties of the set of α-differential mappings are discussed and the Galois correspondence theorems set up for α-differential fields.
Section 3 sets up the notion of α-Liouvillian extensions of α-differential fields and briefly discusses the Galois groups associated with these α-Liouvillian extension fields.
Section 4 points to the procedure for the algebraic characterization of ω-simple-α-differential field extensions by elementary solutions of the partial differential equationΔ α ψ m =0.
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References
Ritt, J. F.: Differential algebra. American Mathematical Society Colloquium Publications33 (1950).
Kolchin, E. R.: Picard-vessiot theory of partial differential fields. Proc. Amer. Math. Soc.3, 596–603 (1952).
Nag, B. D., Sayied, A. M.: Electrodynamics of moving media and the theory of the Cerenkov effect. Proc. Roy. Soc. (London), Ser. A235, 544–551 (1956).
Kolchin, E. R.: Extensions of differential fields. Ann. of Math.43, 724–729 (1942). —Algebraic matrix groups and the Picard-Vessiot theory of homogenous ordinary linear differential equations. Ann. of Math.49, 1–42 (1948).
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Sayied, A.M. Galois theory for the family of partial differential equationsΔ α ψ=0. Math Z 114, 22–32 (1970). https://doi.org/10.1007/BF01111866
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DOI: https://doi.org/10.1007/BF01111866