Quintic polynomials and real cyclotomic fields with large class numbers

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 641491"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>641491</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p = 641491</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the class number of the maximal real subfield of the <italic>p</italic>th cyclotomic field is divisible by the prime 1566401. In an appendix we give a characterization of the "simplest" quadratic, cubic and quartic fields.