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On p-class groups of cyclic extensions of prime degree p of certain cyclotomic fields. (English) Zbl 0716.11053

For an odd prime p let \(\zeta =\exp (2\pi i/p)\) and let k be a cyclic extension of \({\mathbb{Q}}(\zeta)\) of degree p. Let \(C_ k\) be the p-class group of k and let \(r_ k\) be the minimum number of generators of \(C_ k^{1-\sigma}\) as a module over Gal(k/\({\mathbb{Q}}(\zeta))\), where \(\sigma\) is a generator of Gal(k/\({\mathbb{Q}}(\zeta))\). The paper shows how likely it is for \(r_ k\) to equal 0,1,2,... when \(p=3,5,7\) and describes the obstacles to generalizing these results to regular primes \(p>7\); viz. the symmetry property of a certain matrix which fails for regular primes \(p>7.\)
The author has published similar results in Proc. Am. Math. Soc. 90, 1-8 (1987; Zbl 0625.12005) for cyclic extensions k of degree p over \({\mathbb{Q}}\) and it is interesting to compare these with the results of the present paper.
Reviewer: A.R.Rajwade

MSC:

11R29 Class numbers, class groups, discriminants

Citations:

Zbl 0625.12005
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References:

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