On Jacobi sums of certain composite orders

Abstract

The Jacobi sums play a fundamental role in the theory of cyclotomy. Some criteria for power residuacity can be expressed in terms of them; e.g., [9]. Formulas for the number of solutions (h, k)e of ges+h + l m get+k (mod/?), 0Ss,í<where p is a primes 1 (mod e) and g is a primitive root of p, can be expressed in terms of the Jacobi sums of order e [12, (2.7)]. 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]. In his final contribution he studied e=9 and 18. (Sign omissions in formulas (37) were noted in [1]; an ambiguous sign in (44) was also resolved.) The last part of this paper is entitled THEORY FOR fae) = 8, e = 15, 16, 20, 24, 30.One relation associated with e= 16 was omitted. The case e = 15 was left with an undetermined sign. Only sketchy discussions were given to e=20 and 24, while e=30 was ignored completely [5].In this paper the sign ambiguity for e=15 is resolved, and complete analyses for e=24 and 30 are given. Progress is also made toward fixing the sign of a fourth root of unity which arises in connection with the Jacobi sums of order 12. Complete treatments of e= 16 and 20 are found in [11] and [13], respectively.The Jacobi sum R(m, n) of order e is defined by Closely related to the Jacobi sum is the Gaussian sum (2) t(s ) = 2 ßs **« kt\ l = exp (2nilp).)c=i

Jacobi sums over finite fields

“…This example completes [7], [17] where the cyclotomic numbers of order 15 are given in terms of the coefficients of J(χ, χ) and a few other variables, but where no Diophantine system is given for the coefficients of J(χ, χ).…”

On the evaluation of Brewer’s character sums

“…<D14(g7) = -14(-l)/D28(21> 14) = 14(-1)/D28 (7,14), by Theorem 4. According to (3.30) with e = 28, both D2g(0, 14) and D2g(7, 14) are coordinates in a quadratic decomposition of p in eight variables.…”

The Jacobi sums of order twenty-two

“…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]). For e = 20, see [l0].…”