On an extension of the Picard-Vessiot theory

Herbert Frederick Kreimer, Jr.

Vol. 15 (1965), No. 1, 191–205

DOI: 10.2140/pjm.1965.15.191

Abstract

In previous papers, the author has extended the Galois correspondences between differential Picard-Vessiot extensions and algebraic matrix groups to Picard-Vessiot extensions of a wider class of fields with operators, the so-called $M$-fields. In this paper, $M$-field extensions which generalize extensions by integrals and by exponentials of integrals are studied.

These fields are found to be simple field extensions and their structure in the case that the extension is algebraic is investigated. Under suitable restrictions on the fields of constants the $M$-Galois groups of these fields are shown to be commutative. Criteria are established for such solution fields to be $P-V$ extensions of $M$-fields of difference and differential type. An extension obtained by a finite sequence of algebraic extensions, extensions by integrals, and extensions by exponentials of integrals, is called a generalized Liouville extension. It is demonstrated that if the connected component of the identity element in the $M$-Galois group of a regular $P-V$ extension is a solvable group, then the $P-V$ extension is a generalized Liouville extension, and if a $P-V$ extension is contained in a generalized Liouville extension then the connected component of the identity element in the $M$-Galois group of the $P-V$ extension is solvable.

Mathematical Subject Classification

Primary: 12.80

Milestones

Received: 21 February 1963

Revised: 17 March 1964

Published: 1 March 1965