Pencils of higher derivations of arbitrary field extensions
HTML articles powered by AMS MathViewer
- by James K. Deveney and John N. Mordeson PDF
- Proc. Amer. Math. Soc. 74 (1979), 205-211 Request permission
Abstract:
Let L be a field of characteristic $p \ne 0$. A subfield K of L is Galois if K is the field of constants of a group of pencils of higher derivations on L. Let $F \supset K$ be Galois subfields of L. Then the group of L over F is a normal subgroup of the group of L over K if and only if $F = K({L^{{p^r}}})$ for some nonnegative integer r. If $L/K$ splits as the tensor product of a purely inseparable extension and a separable extension, then the algebraic closure of K in L, $\bar K$, is also Galois in L. Given K, for every Galois extension L of K, $\bar K$ is also Galois in L if and only if $[K:{K^p}] < \infty$.References
- James K. Deveney and John N. Mordeson, Invariant subgroups of groups of higher derivations, Proc. Amer. Math. Soc. 68 (1978), no. 3, 277–280. MR 476711, DOI 10.1090/S0002-9939-1978-0476711-7
- James K. Deveney and John N. Mordeson, On Galois theory using pencils of higher derivations, Proc. Amer. Math. Soc. 72 (1978), no. 2, 233–238. MR 507314, DOI 10.1090/S0002-9939-1978-0507314-3
- Nickolas Heerema, Higher derivation Galois theory of fields, Trans. Amer. Math. Soc. 265 (1981), no. 1, 169–179. MR 607115, DOI 10.1090/S0002-9947-1981-0607115-6
- 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
- H. F. Kreimer and N. Heerema, Modularity vs. separability for field extensions, Canadian J. Math. 27 (1975), no. 5, 1176–1182. MR 392951, DOI 10.4153/CJM-1975-123-2
- Saunders Mac Lane, Modular fields. I. Separating transcendence bases, Duke Math. J. 5 (1939), no. 2, 372–393. MR 1546131, DOI 10.1215/S0012-7094-39-00532-6
- John N. Mordeson, On a Galois theory for inseparable field extensions, Trans. Amer. Math. Soc. 214 (1975), 337–347. MR 384762, DOI 10.1090/S0002-9947-1975-0384762-8
- John N. Mordeson and Bernard Vinograde, Structure of arbitrary purely inseparable extension fields, Lecture Notes in Mathematics, Vol. 173, Springer-Verlag, Berlin-New York, 1970. MR 0276204
- William C. Waterhouse, The structure of inseparable field extensions, Trans. Amer. Math. Soc. 211 (1975), 39–56. MR 379454, DOI 10.1090/S0002-9947-1975-0379454-5
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 74 (1979), 205-211
- MSC: Primary 12F15
- DOI: https://doi.org/10.1090/S0002-9939-1979-0524286-7
- MathSciNet review: 524286