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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cyclic extensions of $ K(\sqrt{-1})/K$


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
MathSciNet review: 929665
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Abstract: In this paper the height $ {\text{ht}}(L/K)$ of a cyclic $ 2$-extension of a field $ K$ of characteristic $ \ne 2$ is studied. Here $ {\text{ht}}(L/K) \geq n$ means that there is a cyclic extension $ E$ of $ K,E \supset L$, with $ [E:L] = {2^n}$. Necessary and sufficient conditions are given for $ {\text{ht}}(L/K) \geq n$ provided $ K(\sqrt { - 1})$ contains a primitive $ {2^n}$th root of unity. Primary emphasis is placed on the case $ L = K(\sqrt { - 1})$. Suppose $ {\text{ht}}(K(\sqrt { - 1})/K) \geq 1$. It is shown that $ {\text{ht}}(K(\sqrt { - 1})/K) \geq 2$ and if $ K$ is a number field then $ {\text{ht}}(K(\sqrt { - 1})/K) \geq n$ for all $ n$. For each $ n \geq 2$ an example is given of a field $ K$ such that $ {\text{ht}}(K(\sqrt { - 1})/K) \geq n$ but $ {\text{ht}}(K(\sqrt { - 1})/K) \ngeq n + 1$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0929665-2
PII: S 0002-9947(1989)0929665-2
Article copyright: © Copyright 1989 American Mathematical Society