Cyclotomy and trinomial congruences

The Jacobi Sums of Order Twenty-Two

“…In 1935, Dickson published three papers [3], [4], [5] on cyclotomy which analyzed Jacobi sums of various orders. Further analyses were later given by Whiteman (e=10 [14], 12 [IS], 16 [13]), Muskat (e=\S, 24, 30 [9], 14 [8]), Baumert and Fredricksen (e = 9, 18 [l]), and the author (e = 13, 60 [16]).…”

On Jacobi sums of certain composite orders

“…The relation is (3) R(m,n) = T(m)r(n)lr(m+n), provided e does not divide m, n, or m + n [2, (0.6)]. Letf=(p-l)le.…”

“…The first two relations below were derived from (11) by Dickson (who omitted the (-1)' factor from the former): [5, p. 200] (1824) 72(1,6) = (-l)'/3-3r72 (3,6), (1924) R(l,9) = ß~3Ta7R(l,2).…”

“…Now let e = 30. Since (4130) o7R(l, 13) = R(l, 1), a2372(l, 12) = 72 (1,6), the Jacobi sums of order 30 are conjugate to R(\, «), 1 á«á 11, 72 (1,14), 72 (2,3), and Jacobi sums of lower orders.…”

“…In 1935, L. E. Dickson published a series of three papers which reviewed and extended the theory of cyclotomy. In the first he gave relationships between the Jacobi sums of orders e, e^6, e=8, 10, and 12 [3]. He discussed the sums of prime order and of order twice a prime, then treated explicitly e=14 and 22, in the second paper [4].…”