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On divisibility of \(h^+\) by the prime 5. (English) Zbl 0827.11071
Let \(p\) and \(l\) be primes such that \(p= 2l+1\equiv 7\pmod 8\). The author proves that if the order \(\pmod l\) of 5 is \((l-1)/2\), then 5 does not divide the class number of the maximal real subfield of the \(p\)th cyclotomic field. In a previous paper [Rocky Mt. J. Math. 24, No. 4, 1467–1473 (1994; Zbl 0821.11053)] he established the same result for the prime 3. The present proof is based on similar ideas.

MSC:
11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions
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References:
[1] DAVIS D.: Computing the number of totally positive circular units which are squares. J. Number Theory 10 (1978), 1-9. · Zbl 0369.12002
[2] ESTES D. H.: On the parity of the class number of the field of q-th roots of unity. Rocky Mountain J. Math. 19 (1989), 675-681. · Zbl 0703.11052
[3] JAKUBEC S.: On divisibility of class number of real Abehan fields of prime conductor. Abh. Math. Sem. Univ. Hamburg 63 (1993), 67-86. · Zbl 0788.11052
[4] JAKUBEC S.: On divisibility of h+ by the prime 3. Rocky Mountain J. Math. (1992) · Zbl 0821.11053
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