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On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II

  • Humio Ichimura (a1)

Abstract

Let $m={{p}^{e}}$ be a power of a prime number $p$ . We say that a number field $F$ satisfies the property $\left( {{H}^{'}}_{m} \right)$ when for any $a\in {{F}^{\times }}$ , the cyclic extension $F\left( {{\zeta }_{m}},{{a}^{1/m}} \right)/F\left( {{\zeta }_{m}} \right)$ has a normal $p$ -integral basis. We prove that $F$ satisfies $\left( {{H}^{'}}_{m} \right)$ if and only if the natural homomorphism $C{{l}^{'}}_{F}\to C{{l}^{'}}_{K}$ is trivial. Here $K=F\left( {{\zeta }_{m}} \right)$ , and $C{{l}^{'}}_{F}$ denotes the ideal class group of $F$ with respect to the $p$ -integer ring of $F$ .

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References

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[1] Fröhlich, A. and Taylor, M. J., Algebraic Number Theory. Cambridge Studies in Advanced Mathematics 27, Cambridge Univ. Press, Cambridge, 1993.
[2] Ayala, E. J. Gómez, Bases normales d’entiers dans les extensions de Kummer de degré premier. J. Théor. Nombres Bordeaux 6(1994), 95116.
[3] Greither, C., Cyclic Galois Extensions of Commutative Rings. Lecture Notes in Mathematics 1534, Springer-Verlag, Berlin, 1992.
[4] Ichimura, H., Note on the ring of integers of a Kummer extension of prime degree. II. Proc. Japan Acad. Ser A Math. Sci. 77(2001), 2528.
[5] Ichimura, H., Note on the ring of integers of a Kummer extension of prime degree. IV. Proc. Japan Acad. Ser A Math. Sci. 77(2001), 9294.
[6] Ichimura, H., On the ring of integers of a tame Kummer extension over a number field. J. Pure Appl. Algebra 87(2004), 169182.
[7] Ichimura, H., On a theorem of Kawamoto on normal bases of rings of integers. Tokyo J. Math. 27(2004), 527540.
[8] Ichimura, H., On the ring of p-integers of a cyclic p-extension over a number field. To appear in J. Théor. Nombres Bordeaux.
[9] Kawamoto, F., On normal integral bases. Tokyo J. Math. 7(1984), 221231.
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On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II

  • Humio Ichimura (a1)

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