CYCLOTOMY OF ORDER 15 OVER GF (p2), p 4, 11(MOD 15)

. For primes p 4, 11 (mod 15) explicit formulae are obtained for the cyclotomic numbers of order 15 over GF(p2).

The purpose of this paper is to give the first complete determination of the cyclotomic numbers (h, k) e in the case e 15, L 2, p 4, 11 (rood 15) (so that q p2 (mod 15)). The two basic properties of cyclotomic numbers (6.1), (6.2) show that the e 2 225 cyclotomic numbers take on at most 46 values. As we have assumed that p (rood 15) the additional property (ph, Pk)15 (h, k)15 reduces the maximum number of possible different values to 28 in the case p 4 (mod 15) and 29 in the case p 11 (rood 15). It should be remarked that in the remaining case for which p2 =_ 1 (mod 15), namely, p 14 (mod 15), the phenomenom of uniform cyclotomy occurs (see ) and there are just 3 different cyclotomic numbers.
As an application of the formulae for the cyclotomic numbers in the case p 4 (mod 15), we prove the following theorem in 7. THEOREM 8. Let p 4 (mod 15) be a prime. Let y be a generator of GF(p2)*.
This completes the proof of Theorem 3.
Next, we evaluate the required Jacobi sums in the case p 11 (mod 15). We prove THEOREM 4. For p 11 (mod 15), with the notation of It is very easily checked that, in fact, (5.6) holds for > 1 We next establish the relation (h, k) e (ph, pk) e (5.7) which will be needed in 6 as well as to establish property (5.8)   where the 9 x 9 coefficient matrix of determinant -3375 -33 53 The values of the Ci(v) are given by We remark that the value v 6 has also been included in the statement of Theorem 5 as the solution of (5.11) yields an additional property of the Appealing to (5.10) we deduct that which are the second and third rows of (5.11). The fourth and fifth rows of (5.11) are obtained as follows. From (5.2) (with n 5, e 15) we have 14 2 Jq(BBv, 5) BlB(i, v)B 5i I: T(j, v)B 5j, which are the required fourth and fifth equations.
When v we have by Theorem 3 3 + 8TUB 4 4TUB The values of the Ci(v) are given by Cl(V p2 2  PROOF. The first row in the matrix equation (5.38) follows from (5.5) and (5.io).
In the case p z 4(mod 15) there are 3 tables depending upon the value of indg5(mod 3). All 3 tables are given below (see Tables 13a, 13b For cases in class () (resp. (B), resp. (y)) the value of (h,k)15 may be deduced from  where (m q-l) and the dependence upon the generator y or ym of GF(q)* s shown by writing it as a subscript.           AN APPLICATION: PROOF OF THEOREM 8.