Vol. 61, No. 2, 1975

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On Euler’s criterion for quintic nonresidues

Kenneth S. Williams

Vol. 61 (1975), No. 2, 543–550
Abstract

Let p be a prime 1 (mod 5). If 2 is a quintic nonresidue (mod p) then 2p15 α (mod p) for some fifth root of unity α5(1) (mod p). Emma Lehmer has given an explicit expression for α5 in terms of a particular solution of a certain quadratic partition of p. In this paper we show how in principle the corresponding result can be obtained for any quintic nonresidue D (mod p). Full details are given for D = 2,3,5.

Mathematical Subject Classification
Primary: 10A10, 10A10
Milestones
Received: 4 June 1974
Revised: 8 August 1975
Published: 1 December 1975
Authors
Kenneth S. Williams