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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be fields of characteristic . 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 has a finite separating transcendence basis over *K* for some nonnegative integer *e*. If and 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 for some nonnegative integer *r*.

**[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)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
12F15

Retrieve articles in all journals with MSC: 12F15

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