Mathematical Research Letters

Volume 30 (2023)

Number 2

Counting quintic fields with genus number one

Pages: 577 – 588



Kevin J. McGown (Department of Mathematics and Statistics, California State University, Chico, Calif., U.S.A.)

Frank Thorne (Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)

Amanda Tucker (Department of Mathematics, University of Rochester, New York, U.S.A.)


We prove several results concerning genus numbers of quintic fields: we compute the proportion of quintic fields with genus number one; we prove that a positive proportion of quintic fields have arbitrarily large genus number; and we compute the average genus number of quintic fields. All of these results also hold when restricted to $S_5$-quintic fields only.

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Received 23 June 2020

Received revised 9 February 2023

Accepted 7 April 2023

Published 13 September 2023