Quintic polynomials and real cyclotomic fields with large class numbers

Schoof, René; Washington, Lawrence C.

doi:10.1090/S0025-5718-1988-0929552-2

Quintic polynomials and real cyclotomic fields with large class numbers
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by René Schoof and Lawrence C. Washington PDF

Math. Comp. 50 (1988), 543-556 Request permission

Abstract:

We study a family of quintic polynomials discoverd by Emma Lehmer. We show that the roots are fundamental units for the corresponding quintic fields. These fields have large class numbers and several examples are calculated. As a consequence, we show that for the prime $p = 641491$ the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401. In an appendix we give a characterization of the "simplest" quadratic, cubic and quartic fields.

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