The structure of inseparable field extensions

Waterhouse, William C.

doi:10.1090/S0002-9947-1975-0379454-5

The structure of inseparable field extensions
HTML articles powered by AMS MathViewer

by William C. Waterhouse PDF

Trans. Amer. Math. Soc. 211 (1975), 39-56 Request permission

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.

References

Infinite abelian groups

41

Nathan Jacobson, Lectures in abstract algebra. Vol III: Theory of fields and Galois theory, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London-New York, 1964. MR 0172871, DOI 10.1007/978-1-4612-9872-4
Linda Almgren Kime, Purely inseparable, modular extensions of unbounded exponent, Trans. Amer. Math. Soc. 176 (1973), 335–349. MR 311630, DOI 10.1090/S0002-9947-1973-0311630-8
Warren May, Commutative group algebras, Trans. Amer. Math. Soc. 136 (1969), 139–149. MR 233903, DOI 10.1090/S0002-9947-1969-0233903-9
Moss Eisenberg Sweedler, Structure of inseparable extensions, Ann. of Math. (2) 87 (1968), 401–410. MR 223343, DOI 10.2307/1970711
Richard Rasala, Inseparable splitting theory, Trans. Amer. Math. Soc. 162 (1971), 411–448. MR 284421, DOI 10.1090/S0002-9947-1971-0284421-2