Maximal separable subfields

Author:
Bonnie Page Danner

Journal:
Proc. Amer. Math. Soc. **68** (1978), 125-131

MSC:
Primary 12F15

DOI:
https://doi.org/10.1090/S0002-9939-1978-0460300-4

MathSciNet review:
0460300

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If is a finitely generated separable field extension of characteristic and *M* is an intermediate field such that is inseparable, it is proved there exist subfields *S* of *M* maximal with respect to the property that is separable. These maximal separable subfields, denoted *S*-subfields for , are characterized in two ways.

(1) Let be a separable field extension. Then *S* is a *S*-subfield for if and only if and *S* is algebraically closed in *M*.

(2) If is separable, *S* is a *S*-subfield for if and only if the inseparability of is equal to the transcendence degree of .

A *S*-subfield for is constructed using a maximal subset of a relative *p*-basis for which remains *p*-independent in *L*. It is proved that there is a unique *S*-subfield for if and only if is algebraic for some *S*.

**[1]**Jean Dieudonné,*Sur les extensions transcedants separables*, Summa Brasil Math.**2**(1947), 1-20. MR**0025441 (10:5c)****[2]**Nathan Jacobson,*Lectures in abstract algebra*. III, Van Nostrand, Princeton, N.J., 1964. MR**0172871 (30:3087)****[3]**H. Kraft,*Inseparable Korperweiterrungen*, Comment. Math. Helv.**45**(1970), 110-118. MR**0260709 (41:5333)****[4]**Serge Lang,*Algebra*, Addison-Wesley, Reading, Mass., 1965. MR**0197234 (33:5416)****[5]**Saunders Mac Lane,*Modular fields*. I:*Separating transcendency bases*, Duke Math.**5**(1939), 372-396. MR**1546131**

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/S0002-9939-1978-0460300-4

Keywords:
Separable and inseparable field extensions,
*p*-bases

Article copyright:
© Copyright 1978
American Mathematical Society