Galois theory for fields finitely generated

Authors:
Nickolas Heerema and James Deveney

Journal:
Trans. Amer. Math. Soc. **189** (1974), 263-274

MSC:
Primary 12F15; Secondary 12F10

MathSciNet review:
0330124

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let *K* be a field of characteristic . A subgroup *G* of the group of rank *t* higher derivations is Galois if *G* is the group of all *d* in having a given subfield *h* in its field of constants where *K* is finitely generated over *h*. We prove: *G* is Galois if and only if it is the closed group (in the higher derivation topology) generated over *K* by a finite, abelian, independent normal iterative set *F* of higher derivations or equivalently, if and only if it is a closed group generated by a normal subset possessing a dual basis. If the higher derivation topology is discrete. M. Sweedler has shown that, in this case, *h* is a Galois subfield if and only if is finite modular and purely inseparable. Also, the characterization of Galois groups for is closely related to the Galois theory announced by Gerstenhaber and and Zaromp. In the case , a subfield *h* is Galois if and only if is regular. Among the applications made are the following: (1) is the separable algebraic closure of *h* in *K*, and (2) if is algebraically closed, is regular if and only if is modular for .

**[1]**Nickolas Heerema,*Convergent higher derivations on local rings*, Trans. Amer. Math. Soc.**132**(1968), 31–44. MR**0223358**, 10.1090/S0002-9947-1968-0223358-1**[2]**Nickolas Heerema,*Derivations and embeddings of a field in its power series ring. II*, Michigan Math. J.**8**(1961), 129–134. MR**0136601****[3]**Fredric Zerla,*Iterative higher derivations in fields of prime characteristic*, Michigan Math. J.**15**(1968), 407–415. MR**0238821****[4]**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****[5]**André Weil,*Foundations of Algebraic Geometry*, American Mathematical Society Colloquium Publications, vol. 29, American Mathematical Society, New York, 1946. MR**0023093****[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****[7]**Saunders Mac Lane,*Modular fields. I. Separating transcendence bases*, Duke Math. J.**5**(1939), no. 2, 372–393. MR**1546131**, 10.1215/S0012-7094-39-00532-6**[8]**Morris Weisfeld,*Purely inseparable extensions and higher derivations*, Trans. Amer. Math. Soc.**116**(1965), 435–449. MR**0191895**, 10.1090/S0002-9947-1965-0191895-1**[9]**Moss Eisenberg Sweedler,*Structure of inseparable extensions*, Ann. of Math. (2)**87**(1968), 401–410. MR**0223343****[10]**Murray Gerstenhaber and Avigdor Zaromp,*On the Galois theory of purely inseparable field extensions*, Bull. Amer. Math. Soc.**76**(1970), 1011–1014. MR**0266904**, 10.1090/S0002-9904-1970-12535-6**[11]**Jean Dieudonné,*Sur les extensions transcendantes séparables*, Summa Brasil. Math.**2**(1947), no. 1, 1–20 (French). MR**0025441**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
12F15,
12F10

Retrieve articles in all journals with MSC: 12F15, 12F10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0330124-8

Keywords:
Higher derivation,
iterative higher derivation,
dual basis,
Galois group of higher derivations,
independent abelian sets of higher derivations

Article copyright:
© Copyright 1974
American Mathematical Society