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On Galois theory using pencils of higher derivations


Authors: James K. Deveney and John N. Mordeson
Journal: Proc. Amer. Math. Soc. 72 (1978), 233-238
MSC: Primary 12F15
DOI: https://doi.org/10.1090/S0002-9939-1978-0507314-3
MathSciNet review: 507314
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Abstract: Let $ L \supset K$ be fields of characteristic $ p \ne 0$. Assume K is the field of constants of a group of pencils of higher derivations on L, and hence L is modular over K and K is separably algebraically closed in L. Every intermediate field F which is separably algebraically closed in L and over which L is modular is the field of constants of a group of pencils of higher derivations if and only if $ K({L^{{p^e}}})$ has a finite separating transcendence basis over K for some nonnegative integer e. If $ p \ne 2,3$ and $ K({L^{{p^e}}})$ does have a finite separating transcendence basis over K, and F is the field of constants of a group of pencils, then the group of L over F is invariant in the group of L over K if and only if $ F = K({L^{{p^r}}})$ for some nonnegative integer r.


References [Enhancements On Off] (What's this?)

  • [1] R. L. Davis, Higher derivations and field extensions, Trans. Amer. Math. Soc. 180 (1973), 47-52. MR 47 #6664. MR 0318115 (47:6664)
  • [2] J. Deveney, Fields of constants of infinite higher derivations, Proc. Amer. Math. Soc. 41 (1973), 394-398. MR 49 #259. MR 0335478 (49:259)
  • [3] J. Deveney and J. Mordeson, Invariant subgroups of groups of higher derivations, Proc. Amer. Math. Soc. 68 (1978), 277-280. MR 0476711 (57:16270)
  • [4] -, Subfields and invariants of inseparable extensions, Canad. J. Math. 29 (1977), 1304-1311. MR 0472782 (57:12472)
  • [5] N. Heerma, Higher derivation Galois theory of fields (preprint).
  • [6] N. Heerema and D. Tucker, Modular field extensions, Proc. Amer. Math. Soc. 53 (1975), 301-306. MR 0401724 (53:5551)
  • [7] J. Mordeson and B. Vinograde, Structure of arbitrary purely inseparable field extensions, Lecture Notes in Math., vol. 173, Springer-Verlag, Berlin and New York, 1970. MR 43 #1952. MR 0276204 (43:1952)
  • [8] -, Separating p-bases and transcendental extension fields, Proc. Amer. Math. Soc. 31 (1972), 417-422. MR 44 #6655. MR 0289465 (44:6655)
  • [9] W. Waterhouse, The structure of inseparable field extensions, Trans. Amer. Math. Soc. 211 (1975), 39-56. MR 33 #122. MR 0379454 (52:359)
  • [10] M. Weisfeld, Purely inseparable extensions and higher derivations, Trans. Amer. Math. Soc. 116 (1965), 435-449. MR 33 #122. MR 0191895 (33:122)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0507314-3
Keywords: Modular field extension, pencils of higher derivations
Article copyright: © Copyright 1978 American Mathematical Society

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