bibkey: singer1938projective authors: James Singer year: 1938 title: “A theorem in finite projective geometry and some applications to number theory” doi: 10.1090/s0002-9947-1938-1501951-4 url: https://doi.org/10.1090/s0002-9947-1938-1501951-4 claim: “A Singer cycle identifies the points of a finite projective plane with a cyclic group of order p^2+p+1; a line is a (p^2+p+1,p+1,1) cyclic difference set.” strata_touched:

• D5/S3/Geometry/FiniteGeometry/SingerTracePlane license: citation-only triage: anchor

Singer cyclic difference sets

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DOI: 10.1090/s0002-9947-1938-1501951-4

Source: https://doi.org/10.1090/s0002-9947-1938-1501951-4

Crossref identifies James Singer, Transactions of the American Mathematical Society, volume 43 (1938), pages 377–385, with the title above.

The cubic finite-field construction represents a projective line by the kernel of field trace. The kernel has dimension two over the prime field. Multiplication by a non-base scalar gives a distinct plane, meeting the kernel in dimension one. Primitive powers enumerate projective points; therefore each nonzero cyclic difference occurs once. Index doubling permutes the odd-order cycle and preserves these difference multiplicities. This is a formalization of a classical theorem.