## 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.

### MSC:

 11R29 Class numbers, class groups, discriminants

### Keywords:

cyclic extensions of cyclotomic fields; p-class group

Zbl 0625.12005
Full Text:

### References:

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