Complete solution of the cyclotomic problem in $F_q$ for any prime modulus $l, q = p^α, p ≡ 1 (mod l)$

Number of Points on the Projective Curves aYl=bXl+cZl and aY2l=bX2l+cZ2l Defined over Finite Fields, l an Odd Prime

“…Theorem 3 relates the evaluation of Jacobi sums as determined in Theorem 2 to the work done by earlier authors when ord p (mod l )=1 (see [5,1] and the references therein).…”

“…For the case p#1 (mod l ), q= p : #1 (mod l ), the cyclotomic numbers and the Dickson Hurwitz sums of order l over F q have been determined in terms of the coefficients of certain Jacobi sums of order l (see [5,Lemma 5]). We observe that these formulae also hold true for all other cases of ord p (mod l ); the proofs are analogous to the case p#1 (mod l ) considered in [5].…”

“…The cyclotomic problem for prime modulus l, over finite fields F q , q= p : , p#1 (mod l ), has been treated by Gauss (l=2, 3, q= p), Stieltjes (l=3, q= p), Dickson, Whiteman, and Williams (l=3, 5, q= p), Leonard and Williams (l=7, 11, q= p), Hall and Storer (l=3, q= p : ), Parnami, Agrawal, and Rajwade (l 19, q= p : ), and Katre and Rajwade (any l, q= p : ). For the historical background and references, see [5].…”

Cyclotomy and delta units

“…In [11], the map (5.8) was derived from (5.6) and the canonical ordering of the r\¡, but we have changed the normalization of (x, u, v, w) from [10] and [11]. The normalizations (3.1), (4.1), and (5.1) all follow naturally from Jacobi sums; they insure that the character defined by x(g) -Ce coincides with the particular cth-power residue symbol modulo p belonging to the field K [5]. Using Lehmer's u and v with normalized g makes the units translates of ô' instead of ô .…”

Cyclotomy and Delta Units

“…Schoof and Washington showed that Galois action on the quintic translation units (5.6) can be given by (n + 2) + n6-d2 (5)(6)(7)(8) 6^ l + (n + 2)6 ■ When g satisfies (5.7), then (5.6) induces an ordering of the Q¡. The method of Proposition 3.2 can be used to show that with this ordering the image of 0o under (5.8) is 02 when w = 1 , and 63 when w = -1.…”