## 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

### Keywords:

divisibility; cyclotomic fields; class number

Zbl 0821.11053
Full Text:

### 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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