Internat. J. Math. & Math. Sci.
Vol. 9 No. 4 (1986) 665-704
665
CYCLOTOMY OF ORDER 15 OVER GF (p2), p
4, 11(MOD 15)
CHRISTIAN FRIESEN
Departmert of Mathematics and Statistics
University of New Brunswick
Fredericton, New Brunswick, Canada E3B 5A3
JOSEPH B. MUSKAT
Department of Mathematics and Computer Science
Bar-llan University
Ramat-Gan, Israel
BLAIR K. SPEARMAN
Department of Mathematics
College of New Caledonia
Prince George, British Columbia, Canada V2N 1P8
KENNETH S. WILLIAMS
Department of Mathematics and Statistics
Carleton University, Ottawa, Ontario, Canada KIS 5B6
(Received October 16, 1985)
4, 11 (mod 15) explicit formulae are obtained for the
cyclotomic numbers of order 15 over
ABSTRACT.
For primes p
GF(p2).
KEY WORDS AND PHRASES. Cycloomic numbers, Gauss sumsj Jacobi sums, Eisenstein sums.
19B0 MATHEMATICS SUBJECT CLASSIFICATION CODE. 2C20
I.
INTRODUCTION.
Let e 2 and
I. We set q
integers and let p be an odd prime such that e divides
The
ef + I. We fix once and for all a generator y
positive integer f is defined by q
+pZ-1
of the multiplicative group GF(q)*
GF(q) {0} and set g y1+p+
pg
g a 1 be
pg and denote the finite field with q elements by GF(q).
so that g is a primitive root modulo p. For a GF(q)* the index of a with respect
to y, denoted by indy a, is the unique integer m such that a
0 s m _< q 2, and
for a GF(p)* the index of a with respect to g, denoted by
indg a, is the unique
integer n such that a
(mod p), 0 n p 2. These indices are related by the
following congruence- for a GF(p)* we have
ym,
gn
indya
(I + p +
+
p-l) indga
(mod
p
i).
(I.I)
The congruence (1.1) will be used many times throughout the paper.
The number of solutions
GF(q) + GF(q)* {I} of the pair of congruences
ind (a
I) h (mod e)
Y
ind
a
k (mod e)
(I 2)
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
666
is denoted by (h,
I. The numbers
k) e, where h and k are integers such that 0 -< h _< e 1, 0 < k e
(h, k) e are called the cyclotomic numbers of order e over GF(q),
and they depend on p, 2, e and y. It is a central problem in the theory of cyclotomy
to obtain precise formulae for these numbers. This has been done for a number of
values of e and z--for small values of e these formulae are given in [11] and references to further results are given, for example, in [9].
The purpose of this paper is to give the first complete determination of the
15, L 2, p 4, 11 (rood 15) (so that
cyclotomic numbers (h, k) e in the case e
q
(mod 15)). The two basic properties of cyclotomic numbers (6.1), (6.2)
225 cyclotomic numbers take on at most 46 values. As we have
show that the e 2
reduces the
(h,
assumed that p
(rood 15) the additional property (ph,
4 (mod 15) and 29
maximum number of possible different values to 28 in the case p
in the case p
11 (rood 15). It should be remarked that in the remaining case for
=_ 1 (mod 15), namely, p
14 (mod 15), the phenomenom of uniform cyclotomy
which
occurs (see [2- Theorem i]) and there are just 3 different cyclotomic numbers.
In the case p 4 (mod 15), the 28 cyclotomic numbers are expressed (see
AB, B AT, BT, AU, BU,
Tables 13a, 13b, 13c) as linear combinations of
P, 1, A
2
A,
B, T, U are defined in Theorem and satisfy
TU, where the integers
15U
T
2
2
(1.3)
AB + B
A -1 (mod 3)-, B 0 (mod 3),
p A
(1.4)
p T 2 + 15U 2 T -1 (mod 3)
two
solutions
between
the
not
alone
does
be
distinguish
It should
noted that (1.3)
(A, B) and (A B, -B), while (1.4) determines T uniquely but determines U only up to
p2
k)15
Pk)15
p2
p2,
2,
2,
2,
sign.
For the case p 11 (mod 15), the 29 cyclotomic numbers are expressed (see
2
2
2
Tables 14a- 14g) as linear combinations of p2 p, 1, XU, XV, U UV, UW, V VW, W
where the integers X, U, V, W are defined in Theorem 2 and satisfy
2
(1.5)
p X 2 + 5U + 5V 2 + 5W 2 X -1 (mod 5)
2
2
UV U
XW V
Again, it should be noted that (1.5) alone does not distinguish between the four
(X, V, -U,-W) (X, -U, -V, W) (X, -V, U, -W) (see [I0]).
solutions (X, U, V, W)
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) *.
Set g yp+l, so that g is a primitive root (mod p). The quantity
y(q-l)/5 y2(q-l)/5 y3(q-l)/5 + y4(q-l)/5 GF(p) and a unique square root of 5
modulo p is given by
V y(q-l)/5 y2(q-l)/5
y3(q-l)/5 + y4(q-l)/5
(mod p).
(1.6)
Then we have
(1.7)
-U (mod 3),
ind 1/2(1 + V)
g
where U is given by Theorem 1.
In particular, we see that 1/2(I + V) (equivalently 1/2(I )) is a cube modulo
4 (mod 15), if and only if, U 0 (mod 3), where U is given by (1.4).
the prime p
A sketch of a proof of this, using classfield theory, has been given by Aigner [1].
1
This result complements that of Emma Lehmer (see [9: Theorem 2]) for the case p
(mod 1,5-).
CYCLOTOMY OF ORDER 15 OVER GF(p
2.
2)
667
BASIC PROPERTIES OF JACOBI, GAUSS AND EISENSTEIN SUMS.
With the notation of {}I, we define the Jacobi sum
e for integers m and n by
n)
jq(6m
z
GF(q)
Bm
ind
y
Bn)
jq(Bm,
over GF(q) of order
+ n ind
y
(2.1)
where S
exp(2i/e). These sums have the following well-known properties (see, for
example, [11: Lemmas 13-16]):
jq(Bm
Bn):
jq(Bn,
(_l)nfjq(B-m-n
Bm)
Bn)
(2.2)
(2.3)
jq(Bm Bn) jq(B-m B-n) q if e m e n e m + n
(2.4)
jq(Bm, Bn) -I if e m e n or e m e n
(2.5)
jq(Bm Bn) q_ 2 if e m
(2.6)
jq(Bm Bn) (_l)nf+l if e m e n e m + n
(2.7)
Jq(B pro, B pn) Jq(Bm, B n)
1, the mapping which sends B to BJ is an automorphism of the cyclotomic
If (j, e)
field Q(B), which we shall denote by
oj. We have
Bn))
oj(jq(m,
jq(jm,
Bjn)
(2.8)
Gq(Bm),
tr
Closely related to the Jacobi sum is the Gauss sum
Integer m by
Gq(Bm)
where tr
denotes the trace of
m ind y
B
I:
defined for any
(2.9)
GF(q)*
from GF(q) to GF(p), that is
2
-1
+ P + P +
+ P
tr
and {
exp(2i/p). Gauss sums have the following properties (see for example [8:
pp. 3-5] and [3: Theorem 2.12])
m) :-I, elm,
(2.10)
Gq(B
G
2(Bm) p(_l)(p
P
+ 1)/e’
if e
m and e’
(2.11)
P +
where we have written e’ for e/(e, m)
Gq(Bm) Gq( m) (-1)mfq, if e
Gq (B pm) Gq(Bm)
(2.12)
m
(2.13)
In addition, these sums satisfy the Davenport-Hasse relations [4: eqns. (0.8) and
(0.91)] (see, for example, [8" p. 20, Theorem 5.1] and [3" 8])
Gq(Bm(q’l)/(p-l))
(-I) -1G(B m)
where the Gauss sum G(B m) is defined in (2.18), and for e
integer t
ty indyy
(2.14)
xy and an arbitrary
+
t).
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
668
The basic relationship between Gauss and Jacobi sums is (see, for example,
Theorem 2.2].)
Gq(Bm) Gq(13 r’)
Jq (Bm’ Bn)
Gq( m+n
if e
(
( n
m
e
m
indya
e
[3"
(2.16)
( m + n
We also define the Jacobi sum
j(sm, Bn)
z
a GF(p) t
+ n
ind
(I
a)
(2.17)
and the Gauss sum
mind a a
(2.18)
z
y
a GF(p)*
m)
We note that if
(so that q p), then
B n) J(B B n) and
G(Bm). The sums j(Bm, Bn) and G(6m) also have tile properties (2.2), (2.3) with the
right hand side replaced by p, (2.4), (2.5) with the right hand side replaced by p
2, (2.6), (2.7), (2.8), (2.10), (2.12) with the right hand side replaced by (-I) mf p,
(2.13), and (2.16).
Next, we define the Eisenstein sums E(B m) over GF(p2). We set
p-1 mind (l+bx)
G(Sm
Jq(Bm,
E(Bm)
z
B
m,
Gq(B
Y
(2.19)
b=O
where x is any element of GF
such that
(2.20)
a + bx a, b O, 1, 2
GF(p
p
(p+I)/2 .) Clearly, E(Bm) is
and x 2
GF (p). (For example, we could take x
1) runs twice through the square
independent of the choice of x as bx(1 < b _< p
roots of the quadratic non-residues of GF(p). Eisenstein sums have the properties
(see, for example, [3- pp. 377-381])
(2.21)
E(B m) p, if e m
(2.22)
E(Bm) -(-I) (p+l)/e’ if e m and e’ P + I
(2.23)
E(B m) E(B -m) p, if e’ p +
(p2),
2)
E(B pm) E(Bm)
E(B jm) if (j, e)
where, again, we have written e’ for e/(e, m).
oj(E(Bm))
(2.24)
(2.25)
Finally, we remark that the Eisenstein sums are related to the Gauss sums as
follows (see, for example, [3" Theorem 2.23])"
mind 2
Y
G
m) if e’ p + I
(2.26)
E(Bm) B
P
2(Bm)/G(B
DETERMINATION OF EISENSTEIN SUMS OVER GF(p 2) FOR e 15, p 4, 11 (mod 15).
p: and p 4, 11 (mod 15). We
2, q
Throughout {}3 and 4, we take e 15,
2
m)
14) in the case p 4
sums
1,
E(B (m
begin by determining the Eisenstein
3.
(mod 15).
Set g
m
THEOREM 1. Let p 4 (mod 15) be a prime. Let y be a generator of GF(p2) *.
yp+l, B exp(2i/15), and
exp(2i/3) 1/2(-1 + r). Then
-indglO
65
E(B) is an integer of Q(vrzi-) of the form T + U=T, where T and U are
CYCLOTOMY OF ORDER 15 OVER GF(p
2)
669
rational integers, and we have
E(B 4)
E(6)
E(B 7)
E(B 2)
E(BI4)
E(B 11)
E(B 13)
E(B 8)
E(B 3)
m
indglO (T
-indglO
UV-I-)
(3.1)
(T + UZ-)
(3.2)
E(B 12)
E(B 9)
E(B 6)
+
(3.3)
-I
-ind 2
E(B I0)
E(B 5)
m
g
-m
(3.4)
(a + B,)
where
(3.5)
J(B, 8) A B
Moreover, A, B, T, U satisfy (1.3), (1.4), as well as the congruences given in
the following chart.
ind 5
A
B
(mod 3)
(mod 5)
-T
(mod 5)
g
0
PROOF.
0
1
0
2
From (1.1) we have for a
nd a
(I + p)
-T
GF(p)*
nd a (mod
g
Y
so that
a
ind
(3.6)
-ind
g
p2
1)
(3.7)
a (mod 3)
and
ind
From (2.18) and (3.8) we see that
G(B n)
G(Bm)
We begin by proving (3.4).
if m
-
(3.9)
n (mod 3)
We have
ind 2
Y
E(B 5)
(3.8)
0 (mod 5)
a
Y
Gq(B5)/G(B 5)
ndg2
Gq(B5)/G(B 2)
(by (2.26))
(by (3.7) and (3.9))
-indg2 {G(B)}2/G(( 2)
ndg2 J(B, B)
=-m
(by (2.14))
(by (2.16))
-ind 2
g (A + B)
where A and B satisfy (1.3) (see, for example, [7- Prop. 8.3.4]).
Appealing to
(2.25), we obtain E(B 10) E(B5), which completes the proof of (3.4).
We note that (3.3) follows from (2.22), and in view of (2.24) and (2.25), to
complete the proof it suffices to prove
indglO
E(B 2)
E(B)
(3.10)
and
ind I0
E(B)
First, we prove
m
(3.10). Taking t
g
(T + U/15)
1, x
3, y
(3.11)
5 in (2.15) and appealing to
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
670
(2.12), we obtain
ind 5
Y
q2Gq(5)
13
Gq() Gq( 4) Gq(B 7) Gq(B I0) Gq(B
Applying (2.26) and (3.9), we have
indg5 q2G(2)
BIB
E( 7 E(B i0) E(
E( 5
{G(B)} 5 E(B) E(
Multiplying both sides by GB) and appealing to (2.!2), (2.23) and (2.24), we obtain
B4
2 ind g 5
G(B)6 E()2 p2 E((2)2 E(5)2
giving
ndg5
G(I) 3 E(()
+
p E((
2)
E(B 5)
(3.12)
As
G(B) 3 J(B, B) G(B 2) G(B)
we obtain from (3.4) and (3.12)
E(B 2)
p(A + Bm)
indglO
(3.13)
E(B)
we
(3.13),
of
We now determine which sign holds in (3.13). Cubing both sides
+/-
obtain
E(B2)3
E(B) 3
+
and so
Since
(3.14)
E(B 6) +/- E(B 3) (mod 3)
E(B 6) -I, we must have the + sign holding in (3.14), giving (3.10).
Raising (3.10) to the 5th power and reducing modulo 5, we obtain
E(B 3)
{E(B2)} 5
E(B 10)
m
-indglO
{E(B)} 5
-= m -indglO
E(B 5)
(mod 5)
Then, appealing to (3.4), we obtain
-ind 5
g
A +Bm 2
B 0 (mod 5), if
A B (mod 5), if
A 0 (mod 5), if
Next, we prove (3.11). We have
g
ndg5
indg5
ndg5
ind 10
-ind 10
o2(
g
E(B))
From this equation, we see that
-indglO
so that m
g
E(B)
is an integer of
-ind 10
g
E()
m
where p
-indg10
This proves
(3.5)
-ind 10
g
E(B 2)
E(B)
Q(/:T). Hence, we have
1/2(t + uVC-),
where t and u are integers such that 4p
2U, and
2T, u
odd, so t
T 2 + 15U 2
-
(mod 3)
(mod 3)
2 (mod 3)
0
E(B) is invariant under the automorphism
-ind 10
02
(mod 5)
(A + Bin)
which gives
E(B)
(3 11)
t
2
t + 15u
u
2.
T + U/:-
(mod 2)
Clearly, t and u cannot both be
CYCLOTOMY OF ORDER 15 OVER GF(p
2)
671
-I (mod 3). Cubing (3.11), we obtain by (3.3)
We now show that T
T T 3 E(B) 3 E(B 3) -I (mod 3)
Finally, we prove (3.6). Raising (3.11) to the fifth power, we obtain by (3.4)
that is
-
indg2
(A + B)
E(B 5)
_=
-ind 10
-ind 10
T5
g
E() 5 _--
g
T (mod 5)
ind 5
T _--
g
-m
(mod 5)
(A + Bm)
which together with (3.15) gives (3.6).
This completes the proof of Theorem 1.
Next we determine the Eisenstein sums E(Bm) (I
_<
14) in the case p
m
11
(mod 15).
THEOREM 2.
Set g
Then
y,
E(B)
p+l
11 (mod 15) be a prime.
Let p
exp(2i/15), and o
B
E( 4)
E(B II)
B
2
E(B 14)
3
(5 1/2
exp(2xi/5)
F.
i=O
E(( 2)
E(B8)
E( 7)
E(B 13)
0
E(B 3)
2
-indg24
E(B 12)
z
i=O
E(6 I0)
E(B 5)
.
4
E(9)
E(B 6)
i=O
where the integers t
(0
_<
<
+
4
indg2-indg3
0
GF(p2) *.
i(I0+2(5)1/2))/4
Let y be a generator of
indg3
(3.16)
tio
4
I: t
i=O
o2
ti Oi
(3.17)
(3.18)
-I
(3.)9)
ti 02i
(3.20)
4) satisfy
4
-I
t
i=O
and are of the form
-
(3.21)
1)
(4X
to
tl= (-1- X) + U + V +W
t2
t3
t4
.
4
i=O
tio
V
W
(3.23)
U + V
W
(3.24)
(-1
X) + U
(-I
X)
(3.25)
(-I -X) -U -V + W
where (X, U, V, W) is a solution of
P X 2 + 5U 2 + 5V 2 + 5W 2
XW V 2 UV U 2
Moreover, we have
(3.22)
X + Ui(5 +
2(5)1/2
X
1/2 + Vi(5
-1
(mod 5)
2(5) 1/2 1/2 + W(5) 1/2
(3.26)
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
672
and
4
tio2i: X + Vi(5 + 2(5)I) 1/2
}1
i=O
PROOF.
From (1.1), we have for a
(1 + p) nd g a
nd a
GF(p)*
(mod p2
Y
2(5)1/2) 1/2
Ui(5
W(5) 1/2
1)
so that
ind
Y
a
0
2
nd
(3.27)
(mod 3)
and
nd
a
Y
g
From (2.18) and (3.27) we see that
G(Bm) G(B n) if m
We set
A G(6) B
From (3.29) and (3.30) we have
n
G(62
(3.29)
(rood 5)
G(62)
(3.30)
G(66) G(611)
G(67 G(612
G(()
(3.28)
(mod 5)
a
A
(3.31)
B
Next appealing to (2.12), (3.29) and (3.31), we have
G(63 G(68 G(613
G(64 G(69 G(B In
(3.32)
(3.33)
p/B
p/A
Also, from (2.18) and (3.27) we have
G(B 5)
G(B 10)
-I
Next, we determine the values of the Gauss sums Gq(6 i) (0
G(1)
A, B, p and g.
Taking m
Thus, taking n
Next, taking m
Also, taking m
(3.34)
<
14) in terms of
4n in (2.14), we obtain
Gq(B3n) {G(B4n)}2
1, 2, 3, 4 and appealing to (3.31),
Gq(63 pZ/A2
Gq(66)g) p2/B2
B2
Gq(6
Gq(612) A2
5, 10 in (2.11), we obtain
Gq(65) Gq(B I0) p
have
(3.35)
(3.32), (3.33), we obtain
(3.36)
(3.37)
(3.38)
(3.39)
(3.40)
0 in (2.10), we
(3.41)
Gq(1)
Further, from (2.13), we have
Gq(6 llm) Gq(6m)
From the Davenport-Hasse relation (2.15) with x
(appealing to (3.40))
Gq(63) p2
and with x
5, y
3, t
3 ind 3
6
Y
2, we have
6 ind 3
Y
6
Gq(66) p2
(3.42)
5, y
3, t
1, we obtain
Gq(6) gq( 66 Gq( 611
(3.43)
Gq(62) Gq(67) Gq( 612
(3.44)
1, the equation (3.43) becomes
Appealing to (3.36), (3.37), (3.42), with m
-2 indg3
2
2
0
Gq(6)
p2 B2/A
(3.45)
CYCLOTOMY OF ORDER 15 OVER GF(p
(3.37), (3.39), (3.42)with m
and appealing to
-4
2)
673
2, the equation (3.44) becomes
indg3 p4/A2B 2
(3.46)
faking square roots in (3.45) and (3.46), we have
-ind 3
g
p B/A
-+0
(3.47)
gq(6 2)2
6)
Gq(B)
-2 ind 3
Gq(B 2)
g
-+o
p2/A
(3.48)
B
Raising both sides of (3.47) and (3.48) to the fifth power and reducing modulo 5, we
see that the + signs must hold in both equations; that is,
-ind 3
g p B/A
(3.49)
o
The values of the
Gq(B)
-2 indg3
p2 /A B
Gq(( 2) o
remaining Gq(g i) follow imn, ediately from (3.49)
Gq(6)) Gq(B 4) Gq(B II) Gq(B 14)
(3.5o)
and (3,50) as
Gq(B 2) Gq(B 7) Gq(B8) Gq(B 14)
(3.51)
(3.52)
We now appeal to (2.22), (2.26) and the above fomulae for the G(B i) and
Gq(B i)
to obtain
E(B)
E( 4)
E(B II)
E(6 2)
E(B 7)
E(6 8)
We now examine
E(B 3)
g
o
2
e
E(613
o
-2
0
ndg 6
indg6
p B /
A2
P2 / A B 2
2 ind 2
g p B / A2
O
E(B 3)
E(B 12)
E(B 5) E(B I0)
E(B 6)
E(B 9)
E(B3).
2 ind 2
E( 14)
0
ndg2 p2/ A B2
We have
p B / A2
2 / G(B
indg2 G(B4)
3)
2 ind g 2
(by (2.16) and (3.29))
e
J(B B
+
a)
4 ind (1
4 ind a
2 ind 2 p-1
Y
g
Y
z B
=-e
a=2
a)
2 indg2 p-1
indga +
4, 4)
indg(1
B
E
8
a:2
2
0
6)
indg2 R5(1,
2 ind g 2
(in the notation of [5- eqn. (3.1)])
1)
4
7.
i=O
B5(i
1)
0
[5" p. 345]
(3.53)
(3.54)
(3.55)
(3.56)
(3.57)
674
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
4
}:
j:O
4
r.
j=O
4
7,
j:O
B5(.i__
I) e j
2 ind 2
g
B5( j
2 ind 2 I)
g’
t.O j
J
where (see [5" eqn. (9.5)]) for 0
_<
B5(J
j
2
_<
-
(p- I)}
0J
4
1
I)
indg2,
(p
I)
(3.58)
From [5. eqn. (9.6)], we have
4
z
j=O
and [5-
(3.59)
tj
Theorem 8]
5t 0 +
4X, t
+ t
2
4V, t
t3
t4
4U
(3.60)
t 2 + t3
t4
t2
t3 + t4
4W,
where X, U, V, W are integers satisfying (3.26). Solving (3.59) and (3.60) for t
O
t 1, t 2, t 3, t 4, we obtain (3.21)
(3.25). This completes the proof of (3.18).
From (3.53) and (3.5b), we have
2 ind 2
ind 3
g
g E(B
e
E(6)
t
3)
which together with (3.18), gives (3.16).
Applying the automorphism
02 to (3.16), we obtain (3.17), and applying the same
automorphism to (3.18), we obtain (3.20).
Finally, we note that
4
4
z
i:O
ti oi
z (t
i:l
to oi
(-X + U + V + W)e + (-X + U
X(-0-02-03-04)
+ U(O +
V
W)o 2 + (-X
W)o 3 +(-X-U-V+W)o 4
U + V
02 03 04)+V(o-O 2 + 03 04)+W(o 02_03 + {)4),
that is
4
z te
X + Ui(5 +
2(5)1/2) 1/2
+ Vi(5
i=O
as required.
Applying
4
r.
i=O
ti 02i
o
2(5)I) I
+ W(5) 1/2,
(3.61)
2(5)1/2) 1/2
W(5) 1/2
(3.62)
2 to (3.61), we obtain
X + Vi(5 +
2(5)1/2) 1/2
Ui(5
as
o2{i(5 + 2(5)1/2) 1/2)
o2(i(5 2(5)t) 1/2
o2(0 + 02 03 04
02 + 04
-e + o 2
i(5
i(5 +
e3 + e4
2(5)1/2) t
and
o2((5) 1/2
This completes the proof of Theorem 2.
(5) 1/2
6)
6
2(5)1/2) 1/2
08
CYCLOTOMY OF ORDER 15 OVER GF(p
4.
2)
675
DETERMINATION OF JACOBI SUMS FOR e 15 AND p _= 4, 11 (mod 15).
B) (1 m 5),
The Jacobi sums of order 15 over GF(p 2) are conjugate to
5). We first evaluate these for p _= 4 (mod 15). We prove
and
THEOREM 3. For p 4 (mod 15), with the notation of Theorem 1, we have
jq(3,
3)
Jq(Bm,
jq(5,
dq(B, B)
2
Jq(B
(A + Bin) (T + UvC-Ig)
(4.1)
(T + U -/Z) 2
(4.2)
B)
dq(B 3,
Jq(B 4
g
)
P
(.3)
(T + U-) 2
(4.4)
ind 5
jq(5
g
)
J(B 3,
Jq(5,5)
PROOF.
B)
-ind 5
(4.5)
(A + B)(T +
B 3)
p
(4.6)
(4.7)
B) 2
(A +
From Theorem 1, we have
indg5 (T + U--in) G()
Gq() Gq( 4) Gq(B II) Gq(i’14)
-indg5 (T + UC-ir) G(B 2)
Gq(B 2) Gq(B 7) Gq(B 8) Gq(B 13)
Gq(( 3) Gq(B 6) Gq(( 9) Gq(B 12) p
and
G(B) 3
Gq(B 5) Gq(B i0)
p(A + Bm) G(B2) 3
(4.8)
(4.9)
(4.10)
(4.11)
G(B) 2
(4.12)
p(A + Bm 2) G(B) G(B 2) P
(2.16).
(4.7) follow from (4.8) (4.12) and
The values of the Jacobi sums (4.1)
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.
11 (mod 15), with the notation of Theorem 2, we have
4
2 indg2 4
3j
For p
Jq(B,
J
Jq(B 3
{i:OI: tioi} {j=OE t.Bj
o
B)
q(B2 ,6)=-0
2
B):-
(4.14)
P
.
4
i=O
Jq(B 4,
Jq(B 5
indg3
2 ind g 3
-o indg2
B)
2
)
B)
Jq(B 3 B 3)
jq(B5
o
2
4
tie
=0
11
}{ 11
j =0
4
I:
i=O
p
(4.15)
(4.16)
4
ind g 2
85)
tie
P
indg2 indg3
(4.13)
tie
2
tjo3 j
(4.17)
(4.18)
(4.19)
676
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K.S. WILLIAMS
PROOF. Theorem 4 follows from (2.16), (3.36)
(3.40), (3.49) (3.57) and
Theorem 2.
5.
DETERMINATION OF DICKSON-HURWITZ SUMS FOR e 15 AND p 4, 11 (mod 15).
For any integers and v, the Dickson-Hurwitz sum Be(i, v) is defined by
e-1
z (h,
vh) e
(5.1)
Be(i, v) h=O
where the cyclotomic number (h, k) e was defined in (1.2).
Jacobi sums and Dickson-Hurwitz sums are related by the formula
e-1
Bn) (_l)nv f z B e (i v) B ni
i= 0
jq(Bnv
(5.2)
The Dickson-Hurwitz sums have the well-known properties
Be(i, v) Be(i, e v I)
and
f
if e
e
O)
1)
Be(i,
Be(i,
f
ifei
(B.3)
=IL
Taking n
(5.4)
0 in (5.2), we obtain
e-1
z
Be(i,
i=O
v)
q
2
In addition, Whiteman [12" Lemma 1] showed in the case
-I v -I) where vv -1
v)=
Be(i,
that if (v, e)
(mod e)
Be(iV
It is very easily checked that, in fact, (5.6) holds for > 1
We next establish the relation
(h, k) e (ph, pk) e
which will be needed in 6 as well as to establish property (5.8) below.
to be the least positive integer, such that
1 (mode), we have
(ph, pk) e
z 1 #
z 1
1
(5.6)
(5.7)
Defining r
pr
GF(q)
I) ph (mod e)
pk (mod e)
nd (
Y
indy
c
pr-lindy(
pr-1 ngy
z I
GF(q)
1
GF(q)
r-i
ind
ind
(-I) p
Y
r-I
Y
P
GF(q)
I) h (mod e)
k (mod e)
h (mod e)
k (mod e)
ind
ind
z
(P
Y
Y
r-I
I) --h (mod e)
P
r-I
k (mod e)
(h, k) e
GF(q)
nd (1- 1) _= h (mod e)
Y
k (mod e)
ind
1
Y
From (5.7) we see that
Be(Pi,
as
v)
Be(i,
.
(5.8)
v)
e-1
Be(Pi,
v)
h=O
(h, pi
vh)
CYCLOTOMY OF ORDER 15 OVER GF(p
From (5.3), (5.4), (5.6), (5.8),
2)
e-1
r. (pk, pi
vpk)e
k=O
e-1
r. (k,
vk) e
k=O
15, q
we see that for e
677
p2,
the values of all
14) are known once the
O, 1
225 Dickson-Hurwitz sums B15(i, v)(i, v
the e
1, 2, 3, 4, 5)
O, 1, 2, 3, 5, 6, 7, 10, 11; v
values of the 45 sums B15(i, v)(i
va|ues
of the 50 sums
4 (mod 15), and the
have been determined in the case p
have been determined
5)
3,
4,
2,
1,
v
5,
4,
12;
6,
8,
9,
2,
3,
I,
O,
(i
v)
B15(i,
11 (mod 15). This is clear in view of the following" for each
in the case p
2
-=
14)
Bl5(i,
O)
Bl5(i,
Bl5(i,
B15(i,
Bl5(i,
v)
Bl5(i, 14 v) (8 < v -<
B15(13i, 13)= B15(13i, 1)
B15(i, 8)
B15(2i, 2)
7)
6)
14)
(5.9)
and for each v
B15ri,
v)
fB15(4i’ v),
4 (mod 15)
if p
(5.10)
(mod
15)
11
15 (11i, v), if p
We begin by determining the system of linear equations which have the nine
v) (i O, I, 2, 3, 5, 6, 7, 10, 11) as solutions when
Dickson-Hurwitz sums
-=
B15(i,
4 (mod 15).
THEOREM 5. For p 4 (mod 15) and for each integer v, 1 < v < 6, the nine
O, I, 2, 3, 5, 6, 7, 10, 11) are the solutions of
Dickson-Hurwitz sums B15(i, v) (i
the matrix equation
p
(5.11)
C(v)
M B(v)
where the 9 x 9 coefficient matrix of determinant -3375
2
-1
0
1
0
1
0
0
0
is independent of v,
0
2
0
1
0
0
2
0
-1
-2
-2
-I
-1
2
0
-1
2
0
0
0
-1
0
2
-1
0
-1
-1
0
0
0
2
0
-I
2
0
-1
0
I
0
0
2
1
-1
0
1
0
0
-I
0
0
-I
-33 53
2
-1
-2
-2
1
-1
-1
0
(5.12)
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN, K. S. WILLIAMS
678
B(v)
The values
B15(0, v)
B15(1, v)
B15(2, v)
B15(3, v)
B15(5, v)
B15(6, v)
B15(7, v)
BI5(IO, v)
B15(11, v)
of the Ci(v) are given
CI(V p2 2
p
if v
C2(v)
-1
if v
C3(v)
-A 2 + B 2
C4(v)
-2AB + B 2
c5(v)
C6(1)
C8(1)
C6(2)
C8(2)
C6(3)
C6(4)
C8(4)
C6(4)
C8(4)
C6(4)
AT
2AU
(T 2
4TU
p
2)
15U
-4TU,
C9(4)
8TU
0
C6(5)
C8(5)
C6(5)
C9(5)
C6(5)
C8(5)
2)
C9(4)
15U
C7(4)
C(v)
C4(v)
C5(v)
C(v)
C7(v)
C8(v)
C(v)
(5. 3)
Cg(v)
by
(5.14)
1, 2, 3, 6
(5.15)
4, 5
0
if v
1, 4,
if v
if v
2, 3, 5, 6
1, 4
(5.16)
(5.17)
(5.18)
-(T 2
(T 2
Cl(V)
C2(v)
2, 3, 5, 6
4AU
2BU
C7(I)
BT 2AU + BU
6TU
C7(2) 8TU
-4TU
CB(2)
C7(3) 0 C8(3) 0 CB(3)
0
3AU
+ 2BU
15U
+ 6TU
4TU
6TU
(T 2
4TU
if v
2)
(5.]9)
C9(I)
(5.20)
(5.21)
0
C7(4)
if
-8TU
indn5 z 0 (mod 3)
C7(4) =4TU
IBU 2)
C8(4)
2TU
-4TU
Z
(5.22)
(mod 3)
if
indg5
C8(4) -(
if indg5
2
2)
15U
2TU
2 (mod 3)
C7(5) -4AU + 2BU
-BT + 2AU BU, if indg5 0 (mod 3)
2BU, C9(5)
BT 3BU, C7(5) 2AU + 2BU, C8(5) -2AU + 4BU
-AT + BT AU BU, if indg5 1 (mod 3)
(5.23)
AT 3AU BT + 3BU, C7(5) 2AU 4BU
4AU
AT AU + 2BU, if indg5 2 (mod 3)
2BU, C9(5)
2
8TU
6TU, C7(6)
C6(6) (T 15U
(5.24)
C8(6) 4TU, CB(6) -4TU
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 Dickson-AT + 3AU,
-2AU
2)
GF(p2)
CYCLOTOMY OF ORDER 15 OVER
679
Hurwitz sums of order 15, namely,
6)
(5.25)
(5.3), (5.6) and (5.25), we obtain
B15(2i, 2) B15(i, 2)
(5.26)
B15(i,
so that by
PROOF.
2)
BlB(i,
(5.11) follows from (5.5) and
The first row in the matrix equation
(5.10).
The second and third rows are derived as follows.
15), we have
4
14
3i
Jq(B3v,B 3)
where
z
2
S(j, v)
11
i=O
3, e
z S(j v) 53 j
v)B
BIB(i,
From (5.2) (with n
(5.27)
(5 28)
v)
B I 5 (j + 5i
Appealing to (5.10) we deduct that
S(4, v)
02, 03
Thus expressing (5.27) in terms of the basis {1, o,
Jq(B 3v, B 3)
for Q(o) we obtain
2 +
S(l, v))(C)
S(l, v)) + (S(2, v)
(S(O, v)
(5.29)
S(2, v)
S(l, v), S(3, v)
03
(5.3o)
From (2.2), (2.4), (2.6) and Theorem 3 we have
jq(B3v,B3)
p
if v
I, 2, 3, 6
-1
if v
4, 5
(5.3z)
and so
C3(v)
S(1, v)
S(2, v)
C2(v)
S(1, v)
S(O, v)
which are the second and third rows of (5.11). The fourth and fifth rows of
15) we have
5, e
are obtained as follows. From (5.2) (with n
Jq(BBv, 5)
14
=0
BlB(i,
2
v)B 5i
v)B 5j,
I: T(j,
j =0
(5.11)
(5.32)
that is
jq(BBv,
55)
T(2, v))m
T(2, v)) + (T(I, v)
(T(O, v)
where
4
s
T(j, v)
From (2.4), (2.6)
i=O
have
3
we
Theorem
and
B15(J
(A + Bm) 2
dq(BBv, 5)
if v
if v
-1
(5.33)
+ 3i, v)
4
(5.34)
2, 3, 5, 6
From (5.32) and (5.34) we derive
(5.35)
T(O, v)- T(2, v) C4(v), T(I, v)- T(2, v)= CB(V)
which are the required fourth and fifth equations.
The sixth, seventh, eighth, ninth rows of (5.11) are obtained by equating the
5
2
coefficients of 1, B, B and B respectively in
Jq(B v,
14
B)
T.
i=O
BlB(i,
v)B
(5.36)
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
680
7}
after representing both sides in terms of a basis {1, B
B of Q(B) with the
6) given by Theorem 3. The
B)(v 1, 2,
values of the Jacobi sums
Jq(B v,
right-hand side of (5.36) in terms of the basis {I,
7
z W(i, v)
B 7} is
B
i=O
where
B15(O,v) B15(2,v) B15(6,v) + B15(7,v) B15(10,v) + B15(11,v),
BiB(l,v) + BIB(2,v) B15(7,v) B15(ll,v),
W(7,v)
B15(2,v) BiB(3,v) + BIB(6,v) B15(ll,v), (5.37)
-BiB(2,v) + BIB(5,v) + BIB(7,v) BiB(lO,v),
W(O,v)
W(l,v)
W(2,v)
W(B,v)
W(B,v)
w(n,v)
-W(3,v)
O.
we have by Theorem 3
When v
Jq(B,
+ (-2AU
2BU)B 3 + (4AU
Jq(B 2,
B)
2AU + BU)B 5 + (2AU
2BU)B
follows. When v
2 we have
U(-I5)1/2) 2
15U 2)
(T +
2B7))
2BU)B 4
+ (BT
(5.19)
2B 5 +
2BU)B + (2AU + 2BU)B 2
3AU) + (4AU
(AT
from which
3
2B + 4B 4
(A + BB 5)(T + U(-3 + 4B + 2B 2
B)
7,
5
4TUB + 4TUB 7,
from which (5.20) follows. The remaining v can be treated similarly. It should be
((T 2
noted that for v
indg5
6TU) + 8TUB + 4TUB 2
3
4TUB + 8TUB 4
4 and 5 there are three cases to be considered depending upon
(mod 3).
This completes the proof of Theorem 5.
We next determine the system of linear equations which have the ten DicksonO, I, 2, 3, 4, 5, 6, 8, 9, 12) as solutions of the matrix
Hurwitz sums B15(i, v)(i
equation
MB(v)
(5.38)
C(v)
where the 10x10 coefficient matrix of determinant
M
1
1
1
0
0
0
I
0
0
0
is independent of v,
2
-1
0
2
0
0
0
0
-1
0
2
-1
0
0
2
0
I
1
1
2
-I
0
0
0
1
0
1
0
0
-2
-2
-2
-2
0
0
0
-1
2
-1
2
0
0
0
-1
0
0
0
1
0
1
0
0
0
0
1
0
-6075
2
-1
0
0
0
2
0
-1
0
0
1
I
-1
-1
-1
-I
0
0
0
-35 52
1
1
0
0
1
0
-I
-1
-1
-1
(5.39)
CYCLOTOMY OF ORDER 15 GF(p
B(v)
v)
Cz(v!
C2(v)
C3(v)
C4(v)
CB(v)
CB(v)
C7(v)
C8(v)
BIB(g,
v)
Cg(v)
The values of
2
if v
p
Cl(V
C2(v)
681
B15(0,
B15(1,
B15(2,
B15(3,
B5(4,
B15(5,
B15(6,
B15(8,
v)
v)
v)
v)
v)
C(v)
v)
v)
B15(12, v)
the Ci(v) are
p2
2)
(5.40)
Co(V)
given by
(5.41)
1, 4
(5.42)
-1
if v
2, 3, 5,
+ indg2)
3, 4,
Ci(1) Ci(3) Q(I + indg2) Q(3
3, 4, 5, 6,
Ci(2) Q(3 + indg2) Q(4 + 2i + indg2)
C3(4) C3(5) -I, Cj(4) Cj(5) O, j 4, 5, 6,
7, 8, 9, 10,
Ci(1) R(2i + 1 + indg2) R(3 + indg2),
p, if indg3 z 2 (mod 5)
(mod 5)
-p, if indg3 z 3i
7, 8, 9, 10
Ci(2)
5, 6,
(5.43)
(5.44)
(5.45)
(5.46)
(5.47)
O, otherwise
Ci(3)
Q(1 +
indg2 + 3indg3)
C7(4) p, Cj(4) O, j 8, 9,
Ci(5) R(3 + indg2 + 2indg3)
Q(2
+
indg2 + 3indg3),
R(2i +
+
indg2 + 2indg3),
to
Q(1)
Q(2)
Q(3)
Q(4)
2tlt 4 +
t22 + 2tot 4 +
t422 + 2tot 3 +
tl + 2tot 2 +
t32 + 2tot I +
tlt 2 + tlt 3 +
R(O)
to 2
R(1)
t22 tot I tot
t42 + tot I + tot
R(3)
R(4)
+
+
2 +
+
3
2
+
+
tot 3 + tot4 +
t32 + tot 2 + tot4 +
tl
(5.48)
(5.49)
10,
where (the t are specified in Theorem 2)
2 +
Q(O)
R(2)
7, 8, 9, 10,
2t2t 3
2tlt 3
2tlt 2
2t3t 4
2t2t 4
t2t 4 +
tlt 4 +
tlt 3 +
t2t 3 +
tlt 2 +
7, 8, 9, 10,
(5.50)
(5.51)
t3t 4
t3t 4
t2t 3
t2t 4
tlt 4
(5.52)
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
682
and
Q(n)
Q(n + 5)
R(n + 5)
R(n)
for all integers n.
PROOF. The first row in the matrix equation
(5.38) follows from (5.5) and
(5.io).
5, e
The second row is derived as follows. From (5.2) (with n
14
5i
B
BlB(i, v)B
i=O
4
2
z B15(3i + j, V))( J
z
j:O i:O
1
4
z (B15(3i + j, v))- B15(3i + 2, v))) J
z
j=O i=O
v)
v) +
v)
v)
.
jq(B 5v, 5)
15) we have
B15(4, v)
B15(3,
B15(2,
B15(1,
BIB(O,
-B15(5, v) + B15(6, v)- B15(8, v) + B15(9, v) + B1512, v)
C2(v) now follow from (4.19), (2.4) and (2.6).
The values of
The third, fourth, fifth, and sixth rows are derived as follows.
(with n 3, e 15), we have
14
14
j
1 S(j, v)O
(i v)B 3i
j=O
i=O
2
j + 5i, v)
S(j, v)
where
i=O
0
we have
-1
Hence, as
3
j
dq(g3v
B3)
From (5.2)
B15
B15(
02 03,
04
jq(3v,
T(j, v)
where
Appealing to (5.10) we obtain
BIB(O, v)
T(I, v) 2B15(I, v)
T(2, v) 2B15(2, v)
T(3, v) B15(3, v)
of C3(v), C4(v), CB(V),
T(O, v)
B3)
T(j, v)O
(5.53)
j=O
S(4, v)
S(j, v)
2B15(4, v) + 2B15(5, v) B15(9, v)
2B15(4, v) + B15(6, v) B15(9, v)
2B15(4, v) B15(9, v) + B15(12, v)
2B15(4, v) + 2B15(8, v) B15(9, v)
and C6(v) follow from (4.18), (2.2),
The values
(2.4), (2.6),
and (2.7), by equating coefficients of 1, 0,
and
in (5.53).
The seventh, eighth, ninth, tenth rows of (5.38) are obtained by equating the
3
2
coefficients of 1, o, o and o
respectively in
J (B
v,
02
03
B): (B 15(0, v)+
+
+
+
B15(2, v) -B 15(5, v)- B15(12, v))
(B15(2, v) + B15(3, v) B15(8, v) B15(12, v))O
(-BIB(I, v) + B15(2, v) + B15(6, v) -B15(12, v))O 2
(B15(2, v) B15(4, v) + B15(9, v) B15(12, v))O 3,
which was obtained from (5.2) (with e
5, n
1) by expressing the right-hand side
in terms of a basis {1, B,
Q(B)
of
and
then using (5.10) to express it in
B
7}
CYCLOTOMY OF ORDER 15 OVER GF(p
2)
683
Jq(B v,
(92,
-C) 3}.
terms of {1, O,
The values of the Jacobi sums
B)(v, I, 2,
6)
are given by Theorem 4.
This completes the proof of Theorem 6.
Next we use Theorems 5 and 6 to determine the values of the Dickson-Hurwitz
sums. We treat the case p 4 (mod 15) first. Inverting the matrix M given in
(5.12), we obtain from (5.11)
B(v)
TNC(v)
(5.54)
where
4
-1
2
2
-3
-3
2
2
-3
2
2
-I
-1
4
-1
-1
4
-1
2
-1
-1
2
-I -I
2
-1
2
-1
-1
-1
-1
-1
-1
2
2
-1
C1(v)
Substituting the values for
obtain the following tables of values of
v
1, 2, 3, 4, 5).
Table 1"
15(0,1):
15(1,1):
15(2,1):
15(3,1):
15(5,1):
15(6,1):
15(7,1):
15(10,1):
B15(11,1)"
15 B
15 B
15 B
15 B
15B
15B
15 B
15 B
15
p
1
1
1
4
-1
-1
-1
4
1
-i
I
-I
1
1
4
-1
B15(0,2):
B15(1,2):
B15(2,2):
B15(3,2):
B15(5,2):
B15(6,2):
B15(7,2):
B15(10,2):
B15(II,2):
-2
-2
-2
-2
-2
-2
-2
-2
-2
1
1
1
1
I
1
1
2
6
2
2
2
-3
-4
-4
-3
4
-2
3
-6
-2
4
3
-2
-2
-4
-2
(5.55)
8
-4
-2
C9(v) specified i, Theorem
B15(i, v) (i O, 1, 2, 3,
B15(i,
AB
B2
AT
BT
-2
1
2
-4
2
2
2
2
-4
-4
2
1
8
1
-4
-2
1
-2
1
1
1
B15(i,
-2
1
-2
1
1
1
-2
-2
-4
-2
-4
1
2)
p
-2
1
8
1
1
-4
-2
1
T2
U2
4
-4
-1
8
-I
1
-2
-4
-2
1
-4
1
-120
-15
-15
30
60
30
-15
60
-15
-4
-1
4
-1
-I
4
-4
-1
-1
-I
-I
-I
I
AU
BU
15
15
-15
-15
15
15
-15
-15
4(mod 15)
p
-I
-I
5 i,to (5.54) we
5, 6, 7, 10, 11;
I), p= 4(mod 15)
A2
Dickson-Hurwitz sums
p2
15
15
15
15
15
15
15
15
15
-2
-4
-2
1
-4
Dickson-Hurwitz sums
p2
Table 2-
8
TU
30
30
-30
-30
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
684
Table 3-
B15(i,
Dickson-Hurwitz sums
p
15
15
15
15
15
15
15
15
15
Table 4a-
2
’15 B15(0,4)-
-6
B15(1,4)"
B15(2,4)"
B15(3,4)"
B15(5,4)"
B15(6,4)"
B15(7,4)"
B15(I0,4)"
B15(II,4)"
15
15
15
15
15
Table 4b"
I
I
-1
-1
-1
-6
-1
-1
-I
-I
-I
2
-4
2
2
-I
I
Dickson-Hurwitz sums
B15(0,4)"
B15(1,4)"
B15(2,4)"
B15(3,4)"
B15(5.4)"
B15(6,4)"
B15(7,4)B15(I0,4)B15(11,4)"
1
I
-4
-1
-3
-2
-2
-4
-3
AB
-2
-4
12
A2
-6
p2
15
15
15
15
15
15
15
15
15
B15(0,3)B15(1,3)"
B15(2,3)B15(3,3)B15(5,3)B15(6,3)"
B15(7,3)B15(I0,3)"
B15(II,3)-
4(rood 15)
p
Dickson-Hurwitz sums B 15(i,4), p
p
15
15
15
2
3), p
-
4(mod 15),
B2
T2
-i
I
2
4
2
-I
p
-=
I
4
-2
-I
4(mod 15),
A2
AB
B2
T2
-6
-1
-2
I
-1
1
-2
1
-2
1
2
-4
2
2
2
2
-4
-4
2
4
2
-1
-I
-6
-1
-30
-30
-I
-2
-2
-I
-2
1
I
-2
4
-1
2
-8
-I
U2
TU
1
-2
1
-1
-6
-1
O(mod 3)
indg5
-8
2
2
-4
-4
2
B15(i,4),
-I
-I
30
30
120
15
15
-30
-60
-30
15
-60
15
l(mod 3)
indg5
TU
30
-30
30
-30
U2
-60
-30
15
15
-60
15
-30
120
15
CYCLOTOMY OF ORDER 15 OVER GF(p
Table 4c"
p2
B15(0,4)"
B15(1,4)"
B15(2,4)B15(3,4)B15(5,4)"
B15(6,4)B15(7,4)"
B15(10,4)B15(11,4)-
15
15
15
15
15
15
15
15
15
Table 5a"
-6
A2
AB
-2
2
-4
2
2
2
2
-4
-4
2
-I
-I
I
-1
-6
1
-1
1
-2
I
-2
-I
-6
-1
Dickson-Hurwitz sums
p2
B15(0,5)"
B15(1,5 )B15(2,5)"
B15(3,5)"
B15(5,5)"
B15(6,5)"
B15(7,5)"
B15(10,5)"
B15(II,5)-
15
15
15
15
15
15
15
15
15
Table 5b-
15
15
15
15
15
15
15
15
15
B15(i,4),
Dickson-Hurwitz sums
1
1
1
B15(i,5,,
-3
-5
-3
-5
T2
-2
-!
2
-I
-2
1
p
-8
-1
-1
2
4
2
4
-1
-
4(mod 15),
BU
4
-1
2
-15
-15
15
-I
15
-8
-I
-1
4
2
4(mod 15),
1
-8
4
-1
2
-!
-8
-1
-1
4
2
4
I
O(mod 3)
AU
BT
-5
indg5
-60
15
-30
15
120
15
15
-60
-30
BT
AT
1
-30
-30
4
2
1
-3
-5
-3
30
-1
p2
I
I
I
30
-8
-2
U2
TU
-I
Dickson-Hurwitz sums B 15(i,5), p
B15(0,5).
B15(1,5).
B15(2,5).
B15(3,5).
B15(5,5)B15(6,5).
B15(7,5).
B15( 10,5)B15(11,5)"
B2
2(mod 3)
indg5
4
-I
1
685
4(mod 15),
p
AT
-8
2)
15
-15
-15
15
indg5
1(rood 3)
AU
BU
2
-1
15
-i
15
15
-15
4
-I
15
2
-8
-I
-15
-
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
686
Table 5c"
Dickson-Hurwitz sums
B15(i,5),
p2
15
15
15
15
15
15
15
15
15
B15(0,5)B15(1,5)B15(2,5)"
B15(3,5)"
B15(5,5)B15(6,5)B15(7,5)"
B15(10,5)B15(11,5)"
AT
BT
4
2
-8
-I
-I
-1
-1
4
-I
2
4
2
2
-I
-8
I
-3
-5
-3
I
1
1
Next, we treat the case p
(5.39), we obtain from (5.38)
-5
4(mod 15),
p
-8
4
-1
-I
11 (mod 15).
indg5
AU
2(mod 3)
BU
15
-15
15
15
-15
15
Inverting the matrix M given in
5 N C(v)
B(v)
(5.56)
where
1
1
1
1
2
-1
-1
2
-1
-1
2
4
-1
-1
-1
-1
4
-1
-I -I
1
2
2
-1
-1
-1
4
-1
-1
-I
-1
4
-1
-1
-1
-1
-1
4
-1
-1
-I
-1
-1
4
-I
-I -1
-1
-1
-1
4
-1
4
-1
-1
8
1
1
-2
1
-4
-2
-2
-2
-2
-2
-4
-2
8
-2
-2
-4
(5.57)
I
-2
-4
-2
-2
8
-2
-2
-2
8
-2
Substituting the values of the Ci(v) given in Theorem 6 into (5.56), we obtain the
< 4). There will be 61
Dickson-Hurwitz sums in terms of the Q(i) and R(i) (0 <
tables as follows"
v
no. of tables
5
25
25
1
5
(depending upon
(depending upon
(depending upon
indg2 (mod 5))
indg2, indg3 (mod
indg2, indg3 (mod
(depending upon
(indg2
+ 2
5))
5))
indg3)(mod
5))
As we cannot display them all here, we just give one table for each value of v.
CYCLOTOMY OF ORDER 15 OVER GF(p
Dickson-Hurwitz sums
Table 6"
p2
15
15
15
15
15
15
15
15
15
15
B15(0,1)
B15(1,1)"
B15(2,1)"
B15(3,1)
B15(4,1)B15(5,I)"
B15(6,1)"
B15(8,1)
B15(9,I)B15(12,1)Table 7"
1
1
1
-I
2
-I
-I
2
1
-I
2
2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
B15(0,2)B15(1,2)B15(2,2)"
B15(3,2)"
B15(4,2)"
B15(5,2)B15(6,2)"
B15(8,2)"
B15(9,2)
B15(12,2)-
11(mod 15),
-4
1
1
I
I
I
I
indg2
I
1
1
1
-4
1
-4
1
-4
O(mod 5)
R(1) R(2) R(3) R(4)
8
-2
-2
-2
-2
-4
1
8
1
-4
-2
1
-2
-4
-2
-2
-2
-2
-4
-4
I
1
-2
1
I
-2
1
-4
-4
I
-4
-4
1
1
Dickson-Hurwitz sums
p2
15
15
15
15
15
15
15
15
15
15
p
687
Q(O) Q(1) Q(2) Q(3) Q(4)R(O)
p
2
-1
B15(i,1),
2)
B15(i,2),
p
2
-1
4
2
-1
-4
1
-4
-1
-I
-4
-I
-I
-I
2
-4
-I
-I
2
-8
-4
-4
-4
1
1
-2
-2
-2
-2
8
-2
11(mod 15),
p
indg2
indg3
Q(O)
Q(1)
Q(2)
1
1
-4
1
-4
1
i
I
1
I
I
1
-4
1
-4
I
1
-4
-4
I
I
8
1
-2
-2
I
-2
8
l(mod 5),
2(mod 5).
Q(3)
Q(4)
-4
I
I
I
I
i
1
1
-4
-4
I
1
-4
I
I
I
688
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
Table 8"
Dickson-Hurwitz sums
B15(i,3),
p
11(rood 15),
O(mod 5),
2(rood 5).
indg2
ind 3 _=
g
p
15
15
15
15
15
15
15
15
15
15
B15(0,3)B15(1,3)B15(2,3)"
B15(3,3)"
B15(4,3)"
B15(5,3)"
B15(6,3)B15(8,3)"
B15(9,3)"
B15(12,3)"
Table 9"
2
1
I
1
p
0
-4
0
0
0
0
0
0
0
0
0
-I
-I
Q(O)
Q(1)
Q(2)
Q(3)
Q(4)
-2
0
0
3
5
-5
3
0
-7
3
3
5
0
3
-5
0
-7
0
-2
3
3
0
0
-12
0
0
3
0
3
3
-7
0
-5
3
0
5
3
0
3
-2
3
-5
5
3
0
0
-2
0
3
-7
-4
-I
-I
-4
-1
-4
-4
B15(i,4),
Dicksor1-Hurwitz sums
p2
15
15
15
15
15
15
15
15
15
15
B15(0,4)B15(1,4)B15(2,4)B15(3,4)"
B15(4,4)B15(5,4)B15(6,4)B15(8,4)B15(9,4)B15(12,4)"
P
1
I
i
I
1
1
10
0
0
0
0
-5
0
0
0
0
-6
-I
-1
-1
-I
-6
-1
-1
-I
-1
p
11(mod 15)
CYCLOTOMY OF ORDER 15 OVER GF(p
Table 10"
15
15
15
15
15
15
15
15
15
15
11(mod 15),
Dickson-Hurwitz sums
B15(i,5),
p2
p
I
R(O)
R(1)
0
1
0
0
-8
0
0
-3
0
-5
-3
0
-3
-3
2
-1
2
-1
-I
-I
2
4
-8
B15(0,5):
B15(1,5):
B15(2,5):
B15(3,5):
B15(4,5):
B15(5,5)B15(6,5):
B15(8,5):
BI5(9,b):
B15(12,5):
1
1
0
0
0
0
0
0
0
2)
p
-I
-I
-I
2
2
4
2
2
-I
-8
2
689
indg3
4(mod 5)
R(2)
R(3)
R(4)
2
2
4
-B
-I
-I
-1
2
-1
4
2
+ 2
dg:
-I
4
2
-1
2
-1
-I
-i
2
-1
2
-8
-8
-1
2
2
-I
2
2
GF(p2).
EVALUATION OF CYCLOTOMIC NUMBERS (h,k)15 OVER
The cyclotomic numbers have the following well-know, properties (see, for
example, [11-p. 25])
6.
(h,k)e
(h,k)e
(6.1)
(e-h,k-h)e
(k,h) e
(k +
if f is even,
(.2)
1/2 e, h + 1/2 e) e,
if f is odd,
as well as the property
(h, k) e
(ph, pk) e
p2,
noted in (5.7). Taking e
2, q
15,
p 4, 11(mod 15), and appealing to
(5.7), (6.1) and (6.2), we see that each of the e 2 225 cyclotomic numbers (h,
is equal to one of the 28 cyclotomic numbers
(0,
k
O, I, 2, 3, 5, 6, 7, 10, II
(1,
k
2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13
(6.3)
k
(2, k)i5
5, 7, 8, g, 12
(3,
k:6, 9
k)15
k)I5
k)15
k)5
when p
when p
(5, 10)15
4 (mod 15), and to
(0, k)15
(I, k)15
(2, k)I5
(3, k)l
(5, 10)15
11 (mod 15).
one of the 29
k
O, I, 2,
k
2, 3, 4,
k
5, 7, 8,
k
6, 9
cyclotomic numbers
3, 4, 5, 6, B, 9, 12
5, 6, 7, 8, 9, I0, 11, 13
10, 12
(6.4)
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
690
These relationships are exhibited in Tables 11 and 12, in which (h, k)15 is in
row h(O < h <- 14) and column k(O -< k
14), and we have written A for 10, B for 11, C
for 12, and D for 13.
Table 11"
Relationships between cyclotomic numbers
0
3
2
0 0,0 0,1 0,2 0,3
0,1 O,B 1,2 1,3
2 0,2 1,2 0,7 1,D
3 0,3 1,3 1,D 0,3
4 0,1 1,4 1,8 1,C
5 0,5 1,5 2,5 2,C
6 0,6 1,6 1,7 3,6
7 0,7 1,7 2,7 1,3
8 0,2 1,8 2,8 2,C
9 0,6 1,9 2,9 3,9
10 O,A 1,A 2,7 2,5
11 O,B 1,5 1,9 1,D
12 0,3 I,C 2,C 3,6
13 0,7 I,D 1,8 2,5
14 O,B 1,2 1,3 1,4
Table 12.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
6
7
0,6 0,7
1,6 1,7
1,7 2,7
3,6 1,3
1.9 1,D
1,A 2,7
0,6 1,9
1,9 0,2
2,9 1,8
3,9 2,8
1,6 2,C
1,A 1,2
3,9 2,5
2,8 2,9
1,7 1,8
8
9
0,2 0,6
1,8 1,9
2,8 2,9
2,C 3,9
1,2 1,6
2,5 1,A
2,9 3,9
1,8 2,8
0,7 1,7
1,7 0,6
2,7 1,6
1,3 1,7
1,D 3,6
2,7 1,9
1,9 I,A
over
10
11
O,B
1,5
1,9
1,D
1,2
1,6
1,A
1,2
1,3
1,7
1,5
0,1
1,4
1,8
1,C
O,A
1,A
2,7
2,5
1,A
5,A
1,6
2,C
2,7
1,6
0,5
1,5
2,5
2,C
1,5
Relationships between cyclotomic numbers (h ,k)15over
2
0,2
1,2
0,8
1,D
1,8
2,5
1,9
2,7
2,8
2,8
2,A
I,B 1,7
O,C 1,4 2,C
0,8 1,D 1,8
0,4 1,2 1,3
0
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,2
0,8
0,9
0,5
0,1
5
4
0,1 0,5
1,4 1,5
1,8 2,5
1,C 2,C
O,B 1,5
1,5 O,A
1,9 1,A
I,D 2,7
1,2 2,5
1,6 1,A
1,A 5,A
1,2 1,6
1,3 2,C
1,7 2,7
1,5 1,6
(h,k)15
1
0,1
0,4
1,2
1,3
1,4
1,5
1,6
1,7
1,8
1,9
1,A
3
5
4
0,3 0,4 0,5
1,3 1,4 1,5
1,D 1,8 2,5
O,C 1,4 2,C
1,4 0,1 I,B
2,C 1,B 0,5
3,6 1,7 1,A
1,D 1,3 2,A
2,5 1,2 2,C
3,9 1,A 1,6
2,C 1,6 5,A
1,3 1,2 I,A
3,6 I,D 2,5
2,5 1,9 2,7
1,4 1,5 1,6
6
0,6
1,6
1,9
3,6
1,7
1,A
0,9
1,9
2,8
3,9
1,A
1,6
3,9
2,8
1,7
7
0,2
1,7
2,7
1,D
1,3
2,A
1,9
0,8
1,8
2,8
2,5
1,2
2,C
2,8
1,8
8
9
O,8 0,9
1,8 1,9
2,8 2,8
2,5 3,9
1,2 I,A
2,C 1,6
2,8 3,9
1,8 2,8
0,2 1,7
1,7 0,6
2,7 1,6
I,D 1,9
1,3 3,6
2,A 1,7
1,9 1,A
10
0,5
1,A
2,A
2,C
1,6
5,A
1,A
2,5
2,7
1,6
0,5
1,5
2,5
2,C
1,B
11
0,1
1,B
1,7
1,3
1,2
1,A
1,6
1,2
1,D
1,9
1,5
0,4
1,4
1,8
1,4
GF(p2),p
12
0,3
1,C
2,C
3,6
1,3
2,C
3,9
2,5
I,D
3,6
2,5
1,4
0,3
1,3
I,D
4(mod 15)
13
0,7
14
O,B
I.D 1,2
1,8 1,3
2,5 1,4
1,7 1,5
2.7 1,6
2,8 1,7
2,9 1,8
2,7 1,9
1,9 1,A
2,C 1,5
1,8 1,C
1,3 I,D
0,2 1,2
1,2 0,1
GF(p2),p--11(mod
12
O,C
1,4
2,C
3,6
1,D
2,5
3,9
2,C
1,3
3,6
2,5
1,4
0,3
1,3
1,D
13
0,8
I,D
1,8
2,5
1,9
2,7
2,8
2,8
2,A
1,7
2,C
1,8
1,3
0,2
1,2
15)
14
0,4
1,2
1,3
1,4
1,5
1,6
1,7
1,8
1,9
1,A
I,B
1,4
1,D
1,2
0,1
In order to determine explicit formulae for the cyclotomic numbers in (6.3) and
(6.4), we express each cyclotomic number (h, k)15 in terms of the Dickson-Hurwitz
sums B15(i, v)(i, v O, 1,
14) and then appeal to the formulae for the
v)
derived
{}5.
in
B15(i,
CYCLOTOMY OF ORDER 15 OVER
GF(p2)
691
Whiteman gave the following formula, which expresses each cyclotomic number of
order e, where e is an odd prime, as a linear combination of certain Dickson-Hurwitz
sums of order e [12: Theorem 1]. We give the formula in the context of GF(q), as
Whiteman’s proof for GF(p) goes over without difficulty to this case. We have, for e
O, 1
an odd prime and r, s
e-l,
e-1
(6 5)
s) e (q 1)(1 e) + e(l a + e r. B e (vr + s v)
v=O
e2(r,
r)
where
r
0 (mod e)
if r
(.6)
0 (mod e)
In Theorem 7, we give the corresponding formula to (6.5), where e is the product of
5,
3, y
15, x
two distinct odd primes x and y. The formula will be used with e
q
p 4 11 (mod 15)
We use the well-known identity
0
if r
p2
y-1
Bx(i,
z
v)
j=O
Be(i
+ jx,
v)
(6.7)
xy
e
to simplify formulae appearing in the proof of Theorem 7, often in the form
y-1
z Be(i + jx, v) Bx(i, v)- Be(i, v)
(6.8)
j=l
We also use a special case of the Ramanujan sum [6: pp. 237-238]
k
k, y
I
if x
k
k, y
if x
-(x-l)
nk
B
k
k, y
-(y-I)
if x
n=l
k,
(x-1)(y-1)
if x
e-lr.,
(6.9)
where the prime (’) indicates that the sunBation variable is restricted to values
coprime with e. We prove
THEOREM 7. Let e xy be the product of two distinct odd primes x and y. Then
e-1 we have
O, 1, 2
for r, s
e2(r,
s) e
q + 1
e (a r +
as +
e-1
e-2
+
z
v=l
ar_ s)
r.
[(e-l)Be(vr+s,v)-(x-1)Bx(vr+s,v)-(y-l)BY (vr+s,v)+u=l
Be(vr+s+u,v)]
(v,e)>1
+
e-2
z
v=l
y-2
+y z
w=l
+
+
e-1
v)-(x-l)B x (vs+r,v)-(y-1)B (vs+r,v)+
[(e-l)Be(vS+r
Y
x-2
By(wr+s,w)-(y-2)(q-2)+ xw=1
(e-1)(Be(rax
e-1
(Be(rax
j=l
B e (vs+r+u v)]
B x (wr+s,w)-(x-2)(q-2)
Be(sax + rby,ax))
j,ax) + Be(sax + rby +
+ sby,ax) +
+ sby +
4(q-1) + 2f(x+y) +
such that
’
u=l
j,
ax))
(x-1)(asy + ary) + (y-1)(arx+ asx)
where a and b are integers
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
692
ax + by
PROOF.
Jacobi sums
(6.10)
e
It is well-known that the cycletomic numbers (r,s) e are related to the
B n) by the formula
jq(Bm,
e-1 e-1
e 2 (r, s) e
z
z
j
(Bm Bn)B-mr-ns
(6.11)
m=O n=O q
Splitting the double sum on the right-hand side into five sums according as
O; m O, n # O; m # O, n O; m # O, n # O, m + n e; m # O, n # O, m + n
m
n
we
obtain,
appealing to (2.4),
e;
#
e-1
j (m Bn)B-mr-ns
(6.12)
e( r + s + r-s +
q+l
e 2 (r s) e
m,n=l q
.
m+n#e
The double sum on the right-hand side of (6.12) will be broken up into six sums S
fol ows"
(n, e)
(m, e)
S
m + n # e
(m, e)
S2 m + n # e
(m, e) (n, e) x
S3 m + n # e
(n, e) y
S4 m + n e
{m, e)
S5
(m, e) x (n, e) y
S6
(m, e) y (n, e) x
First, we evaluate S 1. We have
e-1
eol
z, j (m,(n)( -mr-ns
}i
Sl
n=l
m=l
as
q
(m,e)>l
m+n#e
e-1
e-2
z,
v=l
n=l
(v ,e)>l
e-2
z
v=l
n=l
(v ,el>l
e-2 e-1
e-1
I;’
z
v=l
-n(vr+s)
jq(B nv, in)i
.
e-1
i=O
B e (i v)B n i-vr-s)
(by (5.2))
Be(i,v)
=0
s
e-1
I;’
n=l
Bni_vr_s,,
(v ,e)>l
Appealing to (6.9), we obtain
y-1
e-2
S1
z [(x-l)(y-l)Be(vr+s,v)-(x-l) z
u=l
v=l
Be(vr+s+xu,v)
(v ,e)>l
e-1
x-1
(y-l)
that is
e-2
z
$1 v=l
z B (vr+s+yu,v)
u=1 e
+
z’ B e (vr+s+u,v)]
u=1
e-1
[(e-l)Be(vr+s,v)-(x-l)Bx(vr+s,v)-(y-l)By(vr+s,v)+
(v,e)>1
Similarly, we obtain
;’ B (vr+s+u,v)].
e
u=1
(6.13)
CYCLOTOMY OF ORDER 15 OVER GF(p
v=l
693
e-1
e-2
S2
2)
[(e-1)B e (vs+r
v)-(x-1)Bx(vS+r
v)-(y-1)B (vs+r,v)+ r.’B e (vs+r+u,v)].
Y
u=l
(6.14)
Next, we evaluate S 3. We have
y-1 y-1
-x(tr+us)
xt
$3 z z J q (B Bxu)B
t= u=l
t+u#y
y-2 y-1
z
z
dq(XUW, Bxu)B-xu(wr+s)
w=l u=1
y-2 y-1 e-1
s z z
w=1 u=1 i=O
y-2 e-I
r.
w=1 j=O
Be(i,
y-1
Be(J
1)
(y
w)B xui-wr-s)
Bxuj
s
+ wr + s,w)
(by (5.2)
u=l
.
y-1 e-1
y-2 x-1
z B (yt + wr + s,w)
e
w=l t=O
y-2 x-1
y r. z B e (yt + wr + s,w)
w= t= 0
y-2 ey-2
+ s,w)
z r.
y z
w=l
By(wr
y-2
y z
By(wr
w=l j=O
r. B (j + wr + s w)
r.
e
w=l j=O
YJ
y-1 e-1
s B e (j + wr + s,w)
w=l j--O
(by (6.7))
w)
Be(J,
that is
S3
w=l
+ s,w)
(y
2)(q
2)
(by(5.5)).
(6.15)
Similarly we have
x-2
x z B x (wr + s,w)
w= I
Next we evaluate S 5. We have
e-1 e-1
S4
S5
r.
r.
m=l n=l
jq(Bm,
gn)
(x
2)(q
2)
g-mr-ns
xlm
x-1 y-1
J q (B xt
BYU)B-rxt-sy u
t=l u=l
Now we replace t by iax(mod y) and u by jby(mod x) to obtain
2 -sjby 2
y-1 x-1
g
r.
S5
i=1 j=l
y-1 x-1
ax
r.
z o
i=1 j=1
e-1
jq( Biax2
Bjby2)B-riax
x+jy( Jq(B
,
BbY)B
k(Jq(Bax, BbY)B -rax-sby)
tr(Jq(Bax, BbY)B -rax-sby)
o
k=l
tr(Jq(S aX,
B)B -rax-sby)
(by (2.2) and (6.10))
(6.16)
694
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
e-I
tr(
S
i=0
e-I
tr( z
j=O
i-rax-sby)
ax)B
Be(i,
+ sby+ j,ax)8 j)
Be(rax
e-1
e-i
z
j=O
(x
1)Be(rax
1)(y
BIJ
z
=I
B e(rax + sby + j,ax)
+ sby, ax)
(x
t=1
e-1
x-1
(y
(x
y-I
1) s
Be(rax
+ sby + xt, ax)
1) z B e (rax + sby + yt, ax) + s’ B e (rax + sby + j ax)
t=
j=1
1)(y 1)Be(rax + sby, ax) (x 1)(Bx(rax + sby,ax) B e(rax + sby,ax))
(y
1)(By(rax
S5
(e-
e-I
B e(rax + sby,ax)) + z’ B e (rax + sby + j
j=l
+ sby,ax)
ax)
that is
1)Be(rax
(x
+ sby,ax)
1)Bx(sby,O)
(y
1)By(rax,ax)
e-1
z’ B e (rax + sby + j,ax)
+
(6.17)
j=1
Appealing to (5.4) we have
Bx(sby’O)
gsy
fY
and
By(rax,ax) By(rax,
Similarly we obtain
S6
(e-
l)Be(sax
e
1- by)
By(tax,
e
1)
fx
(x
l)Bx(rby,O)
(y
l)By(sax,ax)
+ rby,ax)
rx"
e-1
+
s’ B (sax + rby + j ax)
(6.18)
j=l e
where as above we have
Bx(rby,O
fy-
ry
and
By(sax,ax)
fx
sx"
This completes the proof of Theorem 7.
Applying Theorem 7 and appealing to the tables of the Dickson-Hurwitz sums
obtained in 5, we obtain formulae for the cyclotomic numbers of order 15 over
p z 4, ll(mod 15).
In the case p z 4(mod 15) there are 3 tables depending upon the value of
3). All 3 tables are given below (see Tables 13a, 13b, 13c). The values
of the cyclotomic numbers are given as linear combinations of
I, A 2 AB B 2 AT BT AU BU T2_I5U 2 TU,
GF(p2),
indg5(mod
p2,
P,
using the following abbreviations
x
y
z
-2A 2 + 2AB + B 2
A 2 4AB + B 2
A 2 + 2AB B 2
(6.19)
CYCLOTOMY OF ORDER 15 OVER GF(p
--
2)
695
In the case p 11 (mod 15) there are 25 tables depending upon the values of
indg2 (mod 5) and indg3(mod 5). The values of the cyclotomic numbers are given as
linear compinations of
2
p, I, XU, XV, U 2
after eliminating the Q(i) and R(i) (0
p
(see (3.21)
4) by means of the following formulae
(3.26), (5.51), 5.52)):
(o)
Q()
Q()
Q()
()
and
UV, UW, V 2 VW, W 2
p
+
p
p
p
p
:-p
R(Z) -p
R(O)
+
u
v
+
xu
+
xu
+
+
xu
+
+
xu
+
xv
+
/
u
v
v
u
+
uv
xv
+
u
+
uv
+
uw
uv
+
uw
xv
4U 2
4V 2
+ XV
/
+
u,
+
u
8W
U
+
/
XU+ V
/
v
v
+
vw
vw.
+
(6.20)
vw
vw
2,
4UV- U, + V, +
/
:-p
uw
xv
R():-p / XU+ UV -U,
R()
uV
/
Vz
V, +
Z,
Z,
(6.21)
UW + V + V, + W
Only 7 of these tables are presented in this paper as the remainder can be
deduced from them as indicated below.
For n= 2, 3, 4 is it convenient to define n* 8, 2, 4 respectively. The 18
cases not covered in the given tables can be divided into 3 classes as follows() ind g 2 O(mod 5), ind g 3 n(mod 5), n 2, 3, 4,
(B) ind g 2
(y)
n(mod 5), ind g 3
m(mod 5), ind g 3
O(mod 5), n
2, 3, 4,
n(mod 5), m 1, 2, 3, 4; n 2, 3, 4,
g
For cases in class () (resp. (B), resp. (y)) the value of (h,k)15 may be deduced
from Table 14b (resp. Table 14c, resp. Table 14d (if m n (mod 5)), Table 14e if m
2n (mod 5)), Table 14f (if m
3n (rood 5)), Table 14g (if m 4n (rood 5)) by looking
up the value of (n’h,
and replacing X, U, V, W by
X, -V, U, -W, if n 2,
X, V, -U, -W, if n 3,
X, -U, -V, W, if n 4.
These transformations are clear in view of the basic properties
ind 2
n’k)15
(m-lh, m-lk)
(h, k)
ind
Y
m
m-lindy:(mod q-l)
m
E m (B n)
Y
Ey(Bm-ln
696
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
ym
GF(q)*
where (m q-l)
and the dependence upon the generator y or
of
s shown
by writing it as a subscript.
Table 13a" Cyclotomic numbers
p _= 4(mod 15), indg5 -_- O(mod 3)
(h,k)15,
2
p
225(0, O)
225(0, 1)
225(0, 2)
225(0, 3)
225(0, 5)
225(0, 6)
225(0, 7)
225(0,10)
225(0,11)
225(1, 2)
225(1, 3)
225(1, 4)
225(I, 5)
225(1, 6)
225(1, 7)
225(1, 8)
225(1, 9)
225(1,10)
225(1,12)
225(1,13)
225(2, 5)
225(2, 7)
225(2, 8)
225(2, 9)
225(2,12)
225(3, 6)
225(3, 9)
225(5,10)
Table 13b"
36
-3
-3
-9
12
-9
-3
12
-3
0
-3
12
0
-3
-3
0
-3
-3
-44
-14
-14
-14
-14
-14
-14
-14
-14
12
I
-3
-3
0
I
I
12
12
-3
6
6
0
I
Cyclotomic numbers
p
225(0, O)
225(0, I)
225(0, 2)
225(0, 3)
225(0, 5)
225(0, 6)
225(0, 7)
225(0,10)
225(0,11)
225(1, 2)
225(I, 3)
225(1, 4)
225(1, 5)
225(1, 6)
225(1, 7)
225(1, 8)
225(1, 9)
225(1,10)
225(1,12)
225(1,13)
225(2, 5)
225(2, 7)
225(2, 8)
225(2, 9)
225(2,12)
225(3, 6)
225(3, 9)
225(5,10)
p
2
1
1
1
1
1
1
I
1
1
1
1
1
1
p
36
-3
-3
-9
12
-9
-3
12
-3
0
-3
12
0
-3
-3
0
-3
-3
12
-3
-3
0
12
12
-3
6
6
0
-44
-14
-14
-14
-14
-14
-14
-14
-14
1
1
1
I
1
1
1
1
1
A ,B
AT
BT
AU
BU
x
y
z
x
z
x
y
y
z
x
y
z
x
y
z
x
y
z
y
z
y
x
y
z
z
x
x
x
48
13
-17
-2
-12
-60
13
4
-5
24
-5
13
-12
4
-8
-2
4
7
-2
4
-8
-2
-11
-2
4
-2
7
0
45
15
-60
0
60
-45
0
-15
-15
60
30
-30
0
30
15
-60
-15
30
-30
0
30
-30
-30
15
-30
30
0
0
-45
30
-15
0
15
45
0
30
-30
-60
60
15
0
-30
30
60
-15
-30
30
0
-15
30
-60
15
30
-30
0
-2
13
-12
-17
7
-2
-2
-8
-2
-2
7
-2
13
-2
-2
-2
-8
-2
-2
13
-2
-2
12
(h,k)15, p
4
-11
10
10
12
4(mod 15),
T 2 15U 2
60
2
-4
-10
-24
-10
2
12
-4
-7
2
-4
8
2
11
-7
2
-4
-13
11
2
8
-13
-4
-4
5
5
-12
AT
BT
AU
BU
T2-15U 2
x
y
z
48
-17
13
-2
-12
-2
-17
-12
13
7
-2
-2
-8
13
-2
7
-2
-2
-2
-2
13
12
13
-26
7
24
7
13
-12
-26
1
-2
4
0
15
45
60
0
-60
-15
0
-45
15
30
-30
30
15
60
-15
-30
0
-30
-60
-15
-30
30
30
0
30
-30
0
0
15
0
-75
0
75
-15
0
0
-45
0
0
-15
-30
0
45
0
0
90
0
30
15
-90
0
0
0
0
0
60
-4
2
x
x
y
z
x
y
z
y
z
y
x
y
z
z
x
x
x
-8
-2
-2
-2
-2
-2
12
-2
4
-2
4
-2
4
-2
1
-2
4
4
-8
-8
-24
0
180
120
60
0
-60
-180
0
-120
-30
-60
60
-60
0
-30
30
60
60
-30
30
0
60
30
-60
-60
30
-30
0
l(mod 3)
indg5
A,B
z
x
y
y
z
x
y
z
TU
-10
12
-10
-4
-24
2
-7
11
-13
8
-4
2
-7
11
2
-4
2
-4
8
-4
-13
2
5
5
-12
TU
0
120
180
-60
0
60
-120
0
-180
30
-30
3O
60
-60
-60
-30
30
0
-60
60
60
-60
60
-30
0
-30
30
0
CYCLOTOMY OF ORDER 15 OVER GF(p
Table 13c-
Cyclotomic numbers
p2
225(0, 0)
225(0, 1)
225(0, 2)
225(0, 3)
225(0, 5)
225(0, 6)
225(0, 7)
225(0,10)
225(0,11)
225(1, 2)
225(1, 3)
225(1, 4)
225(I, 5)
225(1, 6)
225(1, 7)=
225(1, 8)
225(I, 9)
225(1,10)
225(1,12)
225(1,13)
225(2, 5)
225(2, 7)
225(2, 8)
225(2, 9)
225(2,12)
225(3, 6)
225(3, 9)
225(5,10)
p
1
1
I
I
I
Table 14a-
-44
-14
-14
-14
-14
-14
-14
-14
-14
36
-3
-3
-9
12
-9
-3
12
-3
0
-3
12
0
-3
-3
0
-3
-3
12
-3
-3
0
12
12
-3
6
6
0
1
1
1
1
1
Cyclotomic numbers
ind 3
0(mod 5)
(h,k)15,
p
-=
2)
697
4(mod 15),
2(rood 3)
indg5
A,B
AT
BT
AU
BU
T 2 15U 2
x
-24
13
13
-14
-12
-14
13
12
-17
4
7
24
7
-17
-12
4
0
-15
-15
0
0
0
15
0
15
0
0
90
0
30
0
0
0
-30
-90
0
-30
0
90
-90
30
0
0
0
0
-15
30
45
0
-45
15
0
-30
-15
-30
-60
45
-15
30
15
30
15
30
-30
15
-45
-30
60
-15
60
-60
0
24
2
2
y
z
x
z
x
y
y
z
x
y
z
-]2
13
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
x
y
z
x
y
z
y
z
y
x
y
z
z
x
x
x
-2
-2
16
16
48
(h,k)15,
p
-
-2
4
13
4
-2
-11
-2
4
13
-2
4
-11
-8
-8
-24
11(mod 15),
indg2
_=
-16
12
-16
2
12
2
2
2
-13
2
2
2
2
2
2
-13
2
2
2
-13
-13
2
14
14
-48
TU
0
60
60
0
0
0
-60
0
-60
0
0
90
0
-120
0
0
0
120
-90
0
120
0
90
-90
-120
0
0
0
0(mod 5),
g
p2
225(0, 0)
225(0, 1)
225(0, 2)
225(0, 3)
225(0, 4)
225(0, 5)
225(0, 6)
225(0, 8)
225(0, 9)
225(0,12)
225(1, 2)
225(1, 3)
225(I, 4)
225(1, 5)
225(1, 6)
225(1, 7)
225(I, 8)
225(1, 9)
225(1,10)
225(1,11)
225(1,13)
225(2, 5)
225(2, 7)
225(2, 8)
225(2,10)
225(2,12)
225(3, 6)
225(3, g)
225(5,10)
I
1
1
1
I
1
I
I
I
I
I
I
I
I
1
I
I
I
p
-82
13
13
13
13
-37
13
13
13
13
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
98
-44
-14
-14
-14
-14
-14
-14
14
-14
-14
1
1
1
1
1
1
1
1
XU
XV
0
-20
-110
50
20
0
-50
110
50
-50
0
0
0
40
20
0
0
0
-20
-40
0
40
80
0
-80
-40
0
0
0
0
-110
20
-50
110
0
-50
-20
50
50
0
0
0
-80
-40
0
0
0
40
80
0
20
40
0
-40
-20
0
0
0
U2
480
-10
-180
30
-10
180
-70
180
-70
30
-80
-20
40
80
-40
-20
100
-20
-40
80
-20
0
-60
220
-60
0
-20
-20
-240
UV
UW
0
160
-160
-I00
160
0
100
160
100
-100
-40
100
-40
-80
-20
-100
40
-100
-20
-80
100
20
80
40
80
20
-200
200
0
0
-30
-60
-150
30
0
-150
60
150
150
0
-60
0
0
60
-180
0
180
-60
0
60
-120
0
0
0
120
0
0
0
V2
480
-180
-10
-70
-180
180
30
10
30
-70
100
-20
220
-60
0
-20
-80
-20
0
-60
-20
-40
80
40
80
-40
-20
-20
-240
VW
0
60
-30
150
-60
0
-150
30
150
-150
0
-180
0
0
-120
60
0
-60
120
0
180
-60
0
0
0
60
0
0
0
W2
240
-130
-130
140
-130
120
140
130
140
140
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
160
-160
-480
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
698
Table 14b"
Cyclotomic numbers
indg3
p2
225(0, O)
225(0, 1)
225(0, 2)
225(0, 3)
225(0, 4)
225(0, 5)
225(0, 6)
225(0, 8)
225(0,
225(0,12
225(1, 2)
225(I, 3)
225(1, 4)
225(1, 5)
225(1, 6)
225(1, 7)
225(1, 8)
225(1, 9)
225(1,10)
225(1,11)
225(1,13)
225(2, 5)
225(2, 7)
225(2, 8)
225(2,10)
225(2,12)
225(3, 6)
225(3, 9)
225(5,10)
91
I
1
I
11(mod 15),
p
indg2
O(mod 5),
l(mod 5)
p
68
13
13
-37
13
-37
13
-37
13
13
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
23
-2
-2
23
-2
-2
23
(h,k)15,
-44
-14
-14
-14
-14
-14
-14
-14
-14
-14
1
XU
XV
U2
UV
UW
V2
VW
W2
-180
40
50
70
0
0
70
10
-30
-130
-10
50
-10
-20
30
20
-10
-10
-50
0
-10
-40
70
-40
-40
10
20
20
90
60
-30
100
10
-50
0
-90
-20
10
10
20
0
20
-60
40
-40
-30
20
0
100
-30
-20
10
30
-20
-20
60
-40
-30
-180
-90
-140
-30
-130
180
-30
180
170
70
0
120
-60
60
20
0
30
60
0
-40
90
-20
-30
-30
-80
-360
-120
80
-60
240
0
-60
0
-60
140
40
40
-140
0
-60
-80
-80
-20
120
0
160
-100
60
40
-40
-60
40
40
180
180
-230
-60
30
30
0
130
60
30
30
60
40
0
-140
60
120
10
-120
40
60
10
-120
-90
-50
60
120
-20
-120
-90
-60
0
-50
190
-160
180
-210
130
-10
-110
70
-110
70
60
-10
160
-110
-50
-90
-40
70
-20
-50
40
100
-110
40
40
30
-60
160
-30
290
-160
0
-110
-70
-10
90
30
-30
-150
-20
-70
60
30
90
70
80
30
-60
-70
0
-120
110
-60
-60
30
-360
-30
-130
140
-30
120
-60
170
-60
140
-30
70
-30
120
-30
120
-80
-30
-30
120
-80
20
20
70
20
-130
40
-60
-180
-180
120
-180
90
CYCLOTOMY OF ORDER 15 OVER GF(p
Table 14c"
Cyclotomic numbers
indg3
p2
225(0,
225(0,
225(0,
225(0,
225(0,
225(0,
225(0,
225(0,
O)
I)
2)
3)
4)
5)
6)
8)
2251,1
225
225(1, 2)
225(1, 3)
225(1, 4)
225(1, 5)
225(1, 6)
225(I, 7)
225(I, 8)
225(I, 9)
225(1,10)
225(1,11)
225(1,13)
225(2, 5)
225(2, 7)
225(2, 8)
225(2,10)
225(2,12)
225(3, 6)
225(3, 9)
225(5,10)
1
1
1
I
1
I
699
ll(mod 15),
p
indg2
-z l(mod 5),
O(mod 5)
P
-7
-17
-7
13
-17
-7
13
13
13
38
3
-2
18
13
13
-12
-2
18
-17
-2
-17
-7
-2
13
-7
-2
-2
-27
38
(h,k)15,
2)
-44
-14
-14
-14
-14
-14
-14
-14
-14
-14
1
1
1
1
XU
XV
U2
UV
UW
V2
VW
W2
90
-20
-20
-10
50
30
-110
-40
-10
40
-40
-50
-10
10
-40
-10
40
20
10
80
10
-20
-40
70
-50
20
40
-10
0
30
50
0
30
-40
0
-70
90
-70
-120
0
-40
30
50
-10
30
-10
-30
-10
-40
-10
0
0
20
-30
-30
80
30
120
30
130
180
-70
40
0
130
-250
-170
-120
-20
40
10
-110
-110
70
10
-170
70
40
10
0
-100
40
150
-10
-20
130
120
300
-200
0
100
-40
120
100
-80
-100
0
20
60
80
-20
-40
140
0
-100
40
80
-120
120
40
60
-60
-140
0
-100
-240
330
90
-240
-70
0
120
30
-70
30
-120
-20
60
70
30
30
-110
30
-50
-90
-120
30
0
80
120
30
-10
-120
130
-120
-270
180
-120
230
70
-90
30
-40
130
-120
-60
-30
-150
30
-80
90
60
-120
90
-20
150
60
20
-30
-30
20
-120
30
0
-210
20
-20
-110
250
-90
-10
40
-110
40
-60
130
-90
-10
40
-30
40
0
-70
40
-110
100
40
10
-110
-80
40
90
240
-60
-30
20
-60
170
-30
-60
-30
140
-160
20
-30
20
-180
-130
20
-30
-130
120
20
120
20
120
-30
20
-30
-60
140
120
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
700
Table 14d"
Cyclotomic numbers
l(mod 5)
ind 3
g
-=
p2
225(0, O)
225(0, I):
225(0, 2):
225(0, 3)
225(0, 4):
225(0, 5)
225(0, 6)
225(0, 8)
225(0, 9)=
225(0,12)225(I, 2)
225(1, 3)
225( 1, 4)
225(1, 5)=
225(1, 6)
225(I, 7)
225(I, 8)
225(1, 9)
225(1,10)
25(I,11)
225(1,13)
225(2, 5)
225(2, 7)
225(2, 8)
225(2,10)
225(2,12)
225(3, 6)
225(3, 9)
225(5,10)
1
I
I
I
I
1
I
I
p
53
-7
-17
-27
-7
-7
23
3
23
-2
-2
8
13
8
8
28
-7
-17
-22
-7
-7
-2
18
-7
13
3
8
-17
8
-44
-14
-14
-14
-14
-14
-14
-14
-14
-14
I
1
(h,k)Ib,
11(mod 15),
p
indg2
-
l(mod 5),
XU
XV
U2
UV
UW
V2
VW
W2
90
-40
-20
-I0
-90
30
-I0
20
90
40
I0
40
-20
-40
30
-20
40
40
20
30
I0
-20
-40
-20
-50
-10
40
-110
0
30
90
0
30
-60
0
30
-30
30
-120
0
-20
0
0
0
0
40
30
-30
-90
40
150
50
20
-150
20
0
-250
-90
50
200
20
-120
-40
80
-I00
-160
0
110
110
-70
120
80
-60
90
-I0
-90
0
50
60
-180
120
40
20
-160
120
-180
80
20
-80
-40
0
80
120
20
-40
-60
80
-120
20
-60
I00
80
60
-20
-220
120
20
0
-390
-110
-40
I0
I00
120
I0
130
10
-40
40
60
-80
40
-20
40
-60
-50
10
-110
180
-I00
40
-30
-I0
-II0
60
I0
240
-390
160
40
II0
90
-90
I0
-100
-90
160
-50
80
-200
40
-90
-80
20
100
100
90
-I0
-20
80
20
-170
50
-40
I0
60
30
-80
180
-270
150
-90
130
-60
30
-120
-30
80
60
-80
90
-180
20
0
-20
-30
-I0
0
120
-40
-30
-30
80
30
120
-60
70
220
140
-30
-30
-260
-130
-60
40
70
-80
-80
-80
-30
-80
-80
70
70
120
70
-80
20
70
-80
20
40
40
120
G
0
-50
-30
30
-20
30
120
CYCLOTOMY OF ORDER Io
Table 14e"
Cyclotomic numbers
l(mod 5)
nd 3
(h,k)15,
OVEK OF(p
2
701
11(mod 15),
p
indg2
2(mod 5),
g
p2
225(0, O)
225(0, I)
225(0, 2)
225(0, 3)
225(0, 4)
225(0, 5)
225(0, 6)
225(0, 8)
225(0, 9)
225(0,12)
225(1, 2)
225(1, 3)
225(I, 4)
225(1, 5)
225(I, 6)
225(1, 7)
225(1, 8)
225(1, 9)
225(1,10)
225(1,11)
225(1,13)
225(2, 5)
225(2, 7)
225(2, 8)
225(2,10)
225(2,12)
225(3, 6)
225(3, 9)
225(5,10)
I
I
I
p
23
-7
-17
-7
3
-7
-7
-7
18
43
8
3
-22
8
-12
23
-12
-7
8
3
18
-17
8
3
-2
8
3
-22
23
-44
-14
-14
-14
-14
-14
-14
-14
-14
-14
XU
XV
U2
UV
UW
V2
VW
W2
90
-10
-10
90
-60
-60
-110
40
-30
90
10
40
90
-30
50
0
0
10
-30
50
0
30
10
-40
30
-20
-10
0
10
-10
-60
-90
30
180
20
-70
60
-90
30
-70
-120
-70
-90
-70
120
120
-30
-30
-10
0
-90
-30
-160
170
80
50
80
-10
-70
180
-150
300
200
40
100
-120
-120
-100
-200
0
100
0
-100
-60
-40
-60
-60
80
120
80
60
-40
40
-80
20
-80
160
-100
0
-60
-210
60
-20
-10
-80
90
-210
50
40
190
-30
30
-60
120
-50
-90
-30
120
30
70
0
70
20
90
-80
-190
90
-60
-30
330
-70
190
-170
-100
0
-70
-20
80
-170
10
-20
190
-100
140
-110
-20
10
-100
50
-80
40
40
10
40
-80
130
-20
-30
270
30
-210
70
-240
120
-30
100
120
-30
10
-60
70
60
0
130
60
10
-60
-90
0
60
-140
-30
60
-20
-30
-80
-90
-60
170
70
-60
-80
-30
140
-30
40
-260
-80
70
70
20
70
-80
70
70
-130
-80
-80
70
120
-80
-80
-30
40
40
120
-110
4C
0
-10
-20
40
-10
10
-40
10
20
-20
-50
20
10
20
1
1
1
1
-110
80
20
-20
-10
20
-20
-10
40
90
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN AND K. S. WILLIAMS
702
Table 14f"
Cyclotomic numbers
1(mod 5)
ind 3
(h,k)15,
11(mod 15),
p
indg2
3(mod 5),
g
p
225(0, O)
225(0, I)
225(0, 2)
225(0, 3)
225(0, 4)
225(0, 5)
225(0, 6)
225(0, 8)
225(0, 9)
225(0,12)
225(1, 2)
225(1, 3)
225(1, 4)
225(1, 5)
225(1, 6)
225(1, 7)
225(I, 8)
225(1, 9)
225(1,10)
225(1,11)
225(I,13)
225(2, 5)
225(2, 7)
225(2, 8)
225(2,10)
225(2,12)
225(3, 6)
225(3, 9)
225(5,10)
2
1
I
I
I
I
I
1
I
p
-7
3
13
-37
13
-7
-12
-7
13
13
-2
8
-2
18
-2
-2
-22
-2
-12
-2
8
-2
38
8
-2
-22
23
-2
38
-44
-14
-14
-14
-14
-14
-14
-14
-14
-14
1
1
1
1
I
1
1
1
1
1
1
XU
XV
U2
UV
UW
V2
VW
W2
-30
20
-80
-30
30
0
20
30
-30
70
-40
0
20
60
-30
20
0
-10
-10
0
-60
70
0
30
40
-30
-30
20
-120
90
-60
-10
-10
20
30
-60
20
-10
-10
40
-70
40
0
-10
40
-40
-50
0
-40
50
-10
-40
20
80
-10
-10
40
0
210
0
10
210
-60
-90
60
-60
-90
10
20
-30
20
-120
30
140
0
180
-80
80
-220
160
-120
280
-40
-20
-220
-40
40
140
-80
100
-40
40
-100
-80
-80
-80
20
80
-80
80
80
-20
80
0
270
180
150
-30
-60
-90
-180
-180
-30
-30
0
-150
60
0
-30
-120
0
90
0
0
-30
90
0
60
0
30
-30
120
0
150
160
-200
!50
-130
0
100
-30
50
-50
0
-60
-60
40
-70
120
120
30
10
-160
60
10
-120
-90
-20
150
-150
0
60
-90
-60
0
-90
270
-120
60
-90
-90
210
0
0
-120
0
-90
-60
0
-30
150
0
60
-30
0
30
0
90
-90
60
0
540
-80
-130
-60
-130
-30
40
-30
140
140
20
-30
20
-80
20
20
120
20
70
20
-30
20
-180
-30
20
120
-60
-160
-180
-70
0
180
-150
-50
-60
0
40
210
-90
-40
-240
CYCLOTOMY OF ORDER 15 OVER GF(P
Table 14g-
Cyclotomic numbers
ind 3
l(mod 5)
g
p2
225(0, O)
225(0, I)=
225(0, 2)=
225(0, 3)
225(0, 4)
225(0, 5)
225(0, 6)
225(0, 8)
225(0, 9)
225(0,12)
225(I, 2)
225(I, 3)
225(1, 4)
225(I, 5)
225(I, 6)
225(1, 7)
225(1, 8)
225(1, 9)
225(1,10)
225(1,11)
225(1,13)
225(2, 5)
225(2, 7)
225(2, 8)
225(2,10)
225(2,12)
225(3, 6)
225(3, 9)
225(5,10)
I
1
1
I
1
1
I
I
1
p
53
33
-7
-2
-37
-7
-27
-17
23
23
-2
-7
-2
18
-7
13
-7
13
-12
8
-44
-14
-14
-14
-14
-14
-14
-14
-14
-14
-7
8
1
1
1
1
13
8
-7
-2
6
-17
8
1
(h,k)15,
2)
703
ll(mod 15)
p
4(mod 5)
ind 2
g
XU
XV
U2
UV
UW
V2
VW
W2
30
-10
60
80
20
-30
-70
-40
-70
30
30
-50
0
-40
-10
30
-50
-30
20
80
-20
-30
-10
40
30
50
-20
30
-60
-270
60
-30
80
50
0
30
20
-70
30
-30
40
-60
60
-40
30
-20
0
0
20
-20
0
50
10
-30
-10
-20
30
0
-210
-140
90
40
110
0
190
40
-10
-210
50
100
-40
-140
80
50
-20
-100
-20
-40
100
-180
70
-110
-30
70
40
-10
240
60
-120
-240
160
-120
120
60
160
60
60
40
20
-20
0
0
220
20
-80
0
0
20
-60
-140
-40
60
40
-40
-140
-120
150
100
-30
0
10
-120
250
-60
-50
-150
-70
20
80
-20
40
50
80
-40
-80
-20
-160
60
30
-70
30
150
-100
50
-120
-30
-250
20
-280
300
-90
170
20
-30
170
-30
110
0
-40
30
-90
-10
-90
80
60
-40
-10
50
20
50
-10
20
-30
-120
-150
-50
60
-60
-130
-30
-160
270
-30
140
20
-60
-60
20
-30
20
20
-30
20
-30
-130
20
-180
120
-30
20
-30
120
20
-60
140
120
0
-20
90
-50
120
-50
-150
-10
10
140
40
70
50
-110
-70
-140
40
-80
30
90
40
90
-150
100
50
-60
PROOF OF THEOREM 8.
Appealing to the well-known congruence (stated here for GF(q))
AN APPLICATION:
7.
ind (I
y
see for example [9- p
-yvf)
e-1
(q
1)/2 +
499], and taking e
u(u v) e (mod e)
}:
u=
15, q
p2
p
_=
(7 I)
4(mod 15), we obtain, as
(see (1.6))
_y6f (I
(I + )
y
6f )(1
y
3f )- 1 (mod p)
(7.2)
the congruence
14
ind
-) =- }: u((u,6)15- (u,3)15)
Y (1
u=1
(mod 3).
(7.3)
Then, making use of Table 11, we obtain
ind
Y
1/2(I + )
(1,3)15 + (1,4)15 + (1,7)15 + (1,10)15 + (2,5)15 + (2,8)15- (1,6)15
-(1,9)15- (1,12)15- (I,13)15- (2,9)15- (2,12)15 (mod 3) (7.4)
the values given in Tables 13a, 13b, 13c for (1,3)15,
(2,12)15 into
Substituting
(7.4) we obtain
indy
1/2(I + ) 1/2(T- A- B)U
independently of the value of
225{(0,0)15 (0,1)15}
and (1.4), we obtain
glven
(mod 3)
3). Reducing the expressions for
by Tables 13a, 13b, 13c modulo 9, and appealing to (1.3)
indg5(mod
704
C. FRIESEN, J. B. MUSKAT, B. K. SPEARMAN
AND K. S. WILLIAMS
T
Combining (7.5) and (7.6) we get
ndy
A
B
1/2(1 + V)
3U2(mod
U3
9).
(7.6)
U (rood 3),
that is
ndg
1/2(1 + /)
(mod 3),
which completes the proof of Theorem 8.
We remark that Theorem 8 can also be proved similarly to the proof of the
case
p--
8.
(mod 15) [9" pp. 499-501].
CONCLUD NG REMARKS.
During the preparation of this paper, a number of Eisenstein sums, Jacobi sums,
Dickson-Hurwitz sums, cyclotomic numbers, were computed for several values of p.
These computations were performed using the facilities of the Computing
Laboratory of
the Department of Mathematics and Statistics at Car]eton University and
the numerical
values obtained were of considerable use in formulating and checking the
results.
The help of Dr. B. C. Mortimer is acknowledged.
ACKNOWLEDGEMENT.
([) Research supported
by a Natural Sciences and Engineering Research Council Canada
Summer Undergraduate Research Award.
(2) Research supported by Natural Sciences and Engineering Research Council Canada
Operating Grant No. A-7233.
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2.
3.
4.
5.
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