Whilst preparing some talks for a seminar based on P.M. Cohn's (blue book) "Skew Field Constructions" (LMS Lecture Note Series 27, 1977) resp. (red book) "Skew Fields - Theory of General Division Rings" (Encyclopedia of Math. & its Appl. 57, Cambridge 1995) I hit upon an unexpected snag in the proof(s) given of the theorem in the title. To be precise, the second-last line p. 39 of the blue book resp. the seventh-last line, p. 99 of the red book, where the author in both cases maintains that $D_1=D$. Due to lack of finiteness conditions here the Double Centralizer Theorem doesn't apply ... probably I'm just missing something simple. Any help in explaining this point would be greatly appreciated ! Kind regards and thanks in advance, St.
I think I may have found something useful: the usual Double Centralizer Theorem (as say in Herstein, "Noncommutative Rings, Th. 4.3.2, or Rowen, "Ring Theory - Student Edition", Th. 7.1.9) does not apply, as indicated, due to lack of finite-dimensionality over the centre. However, Rieffel's (Double Centralizer) Theorem, as say found in Lam, "A First Course in Noncommutative Rings", Th. 3.11, or Lang "Algebra 3.ed", Th. XVII.5.4, is nearer the mark (by taking the small skew field $D$ as the simple ring involved), but the large skew field of the extension is not an ideal of $D$ ... something is still missing, I'm afraid ... St.