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A Trace Conjecture and Flag-Transitive Affine Planes

✍ Scribed by R.D. Baker; G.L. Ebert; K.H. Leung; Q. Xiang

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
127 KB
Volume
95
Category
Article
ISSN
0097-3165

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✦ Synopsis

For any odd prime power q, all (q 2 &q+1)th roots of unity clearly lie in the extension field F q 6 of the Galois field F q 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 F q . Here the trace is taken from F q 6 to F q . 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.

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