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Quintic polynomials and real cyclotomic fields with large class numbers. (English) Zbl 0649.12007

Let \(p\) be a prime and let \(K_p= \mathbb{Q}(J_p)^+\) be the maximal real subfield of the cyclotomic field \(\mathbb{Q}(J_p)\) of the \(p\)-th roots of unity. Vandiver has conjectured that \(h(K_p)\), the class number of \(K_p\), is relatively prime to \(p\). Vandiver’s conjecture would be “trivially” true if, for all \(p\), the prime divisors of \(h(K_p)\) were all less than \(p\). The authors prove that the latter is not always true by showing that \(h(K_{641491})\) is divisible by the prime 1566401.
They find this particular field by looking at primes \(p\) of the form \(n^4+5n^3+15n^2+25n+25\). These primes were studied by E. Lehmer [Math. Comput. 50, No. 182, 535–541 (1988; Zbl 0652.12004)] and are associated with certain cyclic quintic fields whose groups of units are generated by units that are relatively small. The authors study these fields, some of which are the quintic subfields of \(K_p\), and exhibit primes \(p\) for which \(h(K_p)\) exceeds \(p\). It turns out that the quintic subfield of \(K_{641491}\) has class number equal to 1566401. The authors use some results from the geometry of numbers which they apply to groups of units in cyclic quintic fields.

MSC:

11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions
11R21 Other number fields

Citations:

Zbl 0652.12004
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