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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Cyclic Galois extensions and normal bases
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by C. Greither PDF
Trans. Amer. Math. Soc. 326 (1991), 307-343 Request permission

Abstract:

A Kummer theory is presented which does not need roots of unity in the ground ring. For $R$ commutative with ${p^{ - 1}} \in R$ we study the group of cyclic Galois extensions of fixed degree ${p^n}$ in detail. Our theory is well suited for dealing with cyclic ${p^n}$-extensions of a number field $K$ which are unramified outside $p$. We then consider the group $\operatorname {Gal}({\mathcal {O}_K}[{p^{ - 1}}],{C_{{p^n}}})$ of all such extensions, and its subgroup ${\text {NB}}({\mathcal {O}_K}[{p^{ - 1}}],{C_{{p^n}}})$ of extensions with integral normal basis outside $p$. For the size of the latter we get a simple asymptotic formula $(n \to \infty )$, and the discrepancy between the two groups is in some way measured by the defect $\delta$ in Leopoldt’s conjecture.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 326 (1991), 307-343
  • MSC: Primary 11R23; Secondary 11R32, 13B05
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1014248-8
  • MathSciNet review: 1014248