Abstract
The goal of this paper is to introduce some structural ideas into the hitherto chaotic subject of infinite inseparable field extensions. The basic discovery is that the theory is closely related to the well-developed study of primary abelian groups. This analogy undoubtedly has implications beyond those included here. We consider only modular extensions, which are the inseparable equivalent of galois extensions. §§2 and 3 develop the theory of pure independence, basic subfields, and tensor products of simple extensions. The following sections are devoted to Ulm invariants and their computation; the existence of nonzero invariants of arbitrary index is proved by means of a theorem which furnishes an actual connection between primary groups and inseparable fields. The final section displays some complications in the field extensions not occurring in abelian groups.
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