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Title: A Trace Conjecture and Flag-Transitive Affine Planes
Authors: Baker, R.D.
Ebert, G.L.
Leung, K.H. 
Xiang, Q.
Issue Date: Jul-2001
Citation: Baker, R.D., Ebert, G.L., Leung, K.H., Xiang, Q. (2001-07). A Trace Conjecture and Flag-Transitive Affine Planes. Journal of Combinatorial Theory. Series A 95 (1) : 158-168. ScholarBank@NUS Repository.
Abstract: For any odd prime power q, all (q2-q+1)th roots of unity clearly lie in the extension field Fq6 of the Galois field Fq of q elements. It is easily shown that none of these roots of unity have trace -2, and the only such roots of trace -3 must be primitive cube roots of unity which do not belong to Fq. Here the trace is taken from Fq6 to Fq. Computer based searching verified that indeed -2 and possibly -3 were the only values omitted from the traces of these roots of unity for all odd q≤200. In this paper we show that this fact holds for all odd prime powers q. As an application, all odd order three-dimensional flag-transitive affine planes admitting a cyclic transitive action on the line at infinity are enumerated. © 2001 Academic Press.
Source Title: Journal of Combinatorial Theory. Series A
ISSN: 00973165
DOI: 10.1006/jcta.2000.3158
Appears in Collections:Staff Publications

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