Cyclic extensions of

Authors:
Jón Kr. Arason, Burton Fein, Murray Schacher and Jack Sonn

Journal:
Trans. Amer. Math. Soc. **313** (1989), 843-851

MSC:
Primary 12F10; Secondary 11R20

DOI:
https://doi.org/10.1090/S0002-9947-1989-0929665-2

MathSciNet review:
929665

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper the height of a cyclic -extension of a field of characteristic is studied. Here means that there is a cyclic extension of , with . Necessary and sufficient conditions are given for provided contains a primitive th root of unity. Primary emphasis is placed on the case . Suppose . It is shown that and if is a number field then for all . For each an example is given of a field such that but .

**[1]**A. A. Albert,*Modern higher algebra*, Univ. of Chicago Press, Chicago, 1937.**[2]**E. Artin and J. Tate,*Class field theory*, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR**0223335****[3]**Françoise Bertrandias and Jean-Jacques Payan,*Γ-extensions et invariants cyclotomiques*, Ann. Sci. École Norm. Sup. (4)**5**(1972), 517–543 (French). MR**0337882****[4]**Joseph E. Carroll,*On determining the quadratic subfields of 𝑍₂-extensions of complex quadratic fields*, Compositio Math.**30**(1975), no. 3, 259–271. MR**0374082****[5]**J. E. Carroll and H. Kisilevsky,*Initial layers of 𝑍₁-extensions of complex quadratic fields*, Compositio Math.**32**(1976), no. 2, 157–168. MR**0406970****[6]**Burton Fein, Basil Gordon, and John H. Smith,*On the representation of -1 as a sum of two squares in an algebraic number field*, J. Number Theory**3**(1971), 310–315. MR**0319940**, https://doi.org/10.1016/0022-314X(71)90005-9**[7]**I. Reiner,*Maximal orders*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1975. London Mathematical Society Monographs, No. 5. MR**0393100****[8]**O. F. G. Schilling,*The Theory of Valuations*, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. MR**0043776**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
12F10,
11R20

Retrieve articles in all journals with MSC: 12F10, 11R20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0929665-2

Article copyright:
© Copyright 1989
American Mathematical Society