bussey169
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Reprint of W. H. Bussey's paper 'Galois Field Tables for p^n ≤ 169', read before the American Mathematical Society in September 1905 (Bulletin, Oct. 1905). It explains how each field is built from a primitive irreducible congruence and gives two tables per field, one by powers of a primitive root and one by ordered coefficient symbols. Three worked examples show addition, multiplication and division, and the tables cover fields such as GF[2^3], GF[3^2], GF[5^2], GF[7^2] and GF[2^6]. The table text is largely garbled by OCR.
AI-written summary; may contain errors. This description is approximate.
Extracted text (machine-read; may contain errors)
22 GALOIS FIELD TABLES. [Oct.,
GALOIS FIELD TABLES FOR p* ^ 169.
BY DR. W. H. BUSSEY.
(Read before the American Mathematical Society, September 7, 1905).
EVERY field of a finite number of marks may be represented
as a Galois field of order s=pn9 where pn is a power of a
prime. The GF\_pn~\ is defined uniquely by its order, and is
therefore independent of the particular irreducible congruence
used in its construction. In each of the following tables the
GF[pn~\ is constructed by means of a primitive irreducible
congruence which appears at the top of the table. The
marks of each field are arranged in two tables. In each table
each mark appears as a power of a primitive root i, and also
as a polynomial in i of degree k = n — 1. The coefficients in
this polynomial are integers reduced modulo p. The mark
Aik + JBi*-1 + • • • + Di + E, A =f= 0 is denoted by AB• •. DE,
a symbol consisting of its detached coefficients in order. Zero
coefficients must not be omitted. This is the usual symbol for
a positive integer in the notation of the number system whose
base is p. In the first table the marks are arranged according
to ascending powers of i. In the second table the marks are
arranged so that the symbols AB • • • DE represent the positive
integers in natural order. By means of these two tables it is
possible to perform with ease the operations of addition, sub
traction, multiplication and division, within the field.
For an exposition of the Galois field theory, see Dickson's
Linear Groups, pages 1-54 ; Jordan's Traité des Substitutions,
pages 14-18, pages 156-161; Serret's Algèbre supérieure.
For other references on Galois fields and higher irreducible
congruences, see the preface to Dickson's Linear Groups.
Example 1. Simplify (i7 + ils)(i2 + Si -f 4), i being a primi
tive root of the GF[72].
From the first table for GF\_V], ir = 6i + 1, iw = Si + 3,
i2 = i + 4.
Therefore ir + iVà = 9i + 4 = 2i + 4 (modulo 7) = i6 (by
second table).
Also i2 + Si + 4 = 4i + 8 = 4i + 1 (modulo 7) = i22 (by
second table).
Therefore (i7 + in)(i2 + Si + 4) = i6 - i22 = i2S = hi + 1 (by
first table).
1905.] GALOIS FIELD TABLES. 23
Example 2. Simplify in + 3i79 -, i being a primitive 2»26 - 3»'8 + 2 '
root of the GF[5H].
From the first table for GF[BP], i11 = 3Ï2 + 3», »79 = 3»2 + t
_ƒ_ J »26 = 3» + 1 — »'8 = i70 = 3»'2 + 4» + 4.
Therefore*11 + 3i79 = SP + 3» + 3 (3i2 + *'+ 1) = 12»2 + 6»
+ 3 = 2» + « + 3 (mod. 5) = »'81 (by second table).
Also 2»'26- 3»'8 + 2 = 2(3» + 1) + 3(3»2 + 4» + 4) + 2 = 9»2
+ 18»'+16 = 4»2 + 3» + l (mod. 5) = »18 (by second table).
Therefore
iu + in
2i26-3»8+2-»18
Example 3. Simplify = ^. = »63 = 4i (by first table).
- 10i7 + 4 , i being a primitive »-83 + 2/»117
root of the GF[11*].
From the first table for GF[1V], i88 = 6» + 2, i7 = 9t + 2,
t° = 2t + 3, i3= 3»+ 3.
Also — 10i7 = »'7, because — 10 == 1 modulo 11 ; and 2/»'117
= 2»'3 because »120 = 1.
Therefore i88 - 10t7 + 4 = (6t + 2) + (9* + 2) + 4 = 15»
+ 8 = 4» + 8 (modulo 11) = »'82 (by second table).
Also »» + 2/»117 = (2* + 3) + 2 (3» + 3) = 8» + 9 - »101 (by
second table).
Therefore
i®—l0i7 + 4
~T83 + 2/»117 :
table). = 8» + 9 (by first
GF[V], i3 = i + 1, modulo 2. ix = at' + pi + y.
FIEST TABLE.
A
1
2
3
4
5
6
7 a
1
1
1
1 0
1
0
1
1
1
0 Y
0
0
1
0
1
1
1 SECOND TABLE.
\
7
1
3
2
6
4
5 a
1
1
1
1 0
1
1
0
0
1
1 Y
1
0
1
0
1
0
1
24 GALOIS FIELD TABLES. [Oct.,
GF[&], i* si+1, modulo 3. fr = ai + /?.
FIBST TABLE. SECOND TABLE.
A
1
2
3
4 a
1
1
2 0
0 1 |
1
2 1 A
5
6
7
1 8 a
2
2
1 IB
0
2
2
1 A
8
4
1
2 a
1
1 0
1
2
0
1 A
7
1 5
1 3 1 6 a
1
2
2
2 0
2
0
1
2
#F[2*], t<st + l, modulo 2. tA = ai3 + /W* +yi+a.
FIRST TABLE.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15 a
1
1
1
1
1
1
1
1 0
1
0
1
1
0
1
0
1
1
1
1
0 Y
1
0
0
1
1
0
1
0
1
1
1
1
0
0 8
0
0
0
1
0
0
1
1
0
1
0
1
1
1
1 SECOND TABLE.
A
15
1
4
2
8
5
10
3
14
9
7
6
13
11
12 a 0
1
1
1
1
0
0
0
0
1
1
1
1 y
1
1
0
0
1
1
0
0
1
1
0
0
1
1 8
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
QF[5P], i2 = 2î + 2, modulo 5. iK = ai + p.
FIRST TABLE.
A
1
2
3
4
5
6
7
8
9
10
11
12 a
1
2
1
1
4
3
1
3
3
2 0 1
0
2
4
2
2
3
0
1
2
1
1
4 A
13
14
15
16
17
18
19
20
21
22
23
1 24 a
4
3
4
4
1
2
4
2
2
3 0
0
3
1
3
3
2
0
4
3
4
4
1 SECOND TABLE.
A
24
18
6
12
1
8
4
17
3
19
11
2 a
1
1
1
1
1
2
2
2 0 1
1
2 1 3
4
0
1 1
2
3
4
0
1
2 A
21
22
7
10
9
14
23
13
15
5
16
1 20 a
2
2
3
3
3
3
3
4
4
4
4
4 00.
3
4
0
1
2
3
4
0
1
2
3
4
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26 GALOIS FIELD TABLES. [Oct.,
GF[33], iX^ai* + pi + y.
SECOND TABLE.
A
26
13
1
9
3
14
16
22
2 a
1 0
1
1
1
2
2
2
0 7
1
2
0
1
2
0 i
1 1
2 '
o 1 A
21
12
10
6
11
4
18
7
1 15 a
2 0
0
0
1
1
1
2
2
2
0 y
1
2
0
1
2
0
1
2
o 1 A
25
8
17
20
5
23
24
19 a
2
2
2
2
2
2
2
2 0
0
0
1
1
1
2
2
2 y
1
2
0
1
2
0
1
2
O.F[72], i2 = i + 4, modulo 7. iA = at -f- /?.
FIEST TABLE. SECOND TABLE.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24 a
1
1
5
2
1
2
6
3
3
1
6
3
6
4
2
2
3
4
2
4
5 * 1
0
4
4
6
1
4
1
3
0
5
5
4
3
5
3
2
0
1
1
5
2
1
2
6 A
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
1 48 a
6
6
2
5
6
5
1
4
4
6
1
4
1
3
5
5
4
3
5
3
2 0
0
3
3
1
6
3
6
4
0
2
2
3
4
2
4
5
0
6
6
2
5
6
5
1 A
48
16
8
32
40
24
1
5
38
36
2
11
31
17
18
21
27
6
47
4
9
19
44
13 a
2
2
2
2
2
2
2
3
3
3
3 0
1
2
3 1 4 i
5
6
0
1
2
3
4
5
6
0
1
2
3
4
5
6
o 1
2 1 3 1 A
39
10
| 46
1 33
22
34
15
37
20
43
41
28
23
30
3
45
42
25
7
35
26
12
14
1 29 a
3
3
3
4
4
4
4
4
4
4
5
5
5
5
5
5
5
6
6
6
6
6
6
6 0
4
5
6
0
1
2
3
4
5
6
0
1
2
3
4
5
6
0
1
2
3
4
5
6
1905.] GALOIS FIELD TABLES. 27
(£F[26], i6 = i + 1, modulo 2. ^* = <n5 + /?i4 -f >t» + <«* + « + f.
FIBST TABLE.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32 a
1
1
1
1
1
1
1
1
1
1
1
1
1 J8
1
0
1
1
0
1
0
1
0
1
1
1
1
0
1
0
1
1
1
0 y
1
0
0
1
1
0
0
1
0
1
0
0
1
1
1
1
0
1
0
0
1
1
1
0
0
1 s
1
0
0
0
1
1
0
0
0
1
0
1
0
0
1
1
1
1
0
1
0
0
0
1
1
1
0
0
1
0 e
1
0
0
0
0
1
1
0
0
0
1
0
1
0
0
1
1
1
1
0
1
0
0
0
1
1
1
0
0
1
0
0 ç
o 0
0 I
0
0
1
0
0
0
0
1
1
0
0
0
1
0
1
0
0
1
1
1
1
0
1
0
0
0
1
1
1 1 1 A
33
34
35
36
37
38
39
40
41
42
43
44
i 45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63 a
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 f*
1
0
1
0
1
1
0
1
1
1
0
1
1
0
1
1
0
1
0
1
0
1
1
1
1
1
1
0 Y
0
0
1
0
1
1
0
1
1
1
0
1
1
0
0
1
1
0
1
0
1
0
1
1
1
1
1
1
0
0 8
0
1
0
1
1
0
1
1
1
0
1
1
0
0
1
1
0
1
0
1
0
1
1
1
1
1
1
0
0
0 e
1
0
1
1
0
1
1
1
0
1
1
0
0
1
1
0
1
0
1
0
1
1
1
1
1
1
0
0
0
0 i
0
0
1
0
0
1
0
1
1
0
1
1
1
0
1
1
0
0
1
1
0
1
0
1
0
1
1
1
1
1
1
SECOND TABLE.
A
63
1
6
2
12
7
26
3
32
13
35
8
48 a £ V
1
1
1
1
1
1 8
1
1
1
1
0
0
0
0
1
1 6
1
1
0
0
1
1
0
0
1
1
0
0 i
1
0
1
0
1
0
1
0
1
0
1
0
1 A
27
18
4
24
33
16
14
52
36
54
9
45
1 49 a P
1
1
1
1
1
1
1
1
1
1
1 y
1
1
0
0
0
0
0
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1
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1
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30 GALOIS FIELD TABLES. [Oct.,
GF[34], i* = 2i3 + 2*2 + i + 1, modulo 3.
SECOND TABLE.—Continued.
A
50
20
43
16
78
19
47
36
6
55
14
34
8
9
70 a
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2 J8
2
2
0
0
0
0
0
0
0
0
0
1
1
1
1 y
2
2
0
0
0
1
1
1
2
2
2
0
0
0
1 8
1
2
0
1
2 0
1
2
0
1
2
0 1
2
0 1 A
60
10
13
77
35
28
57
61
23
4
11
25
39
71 a
2
2
2
2
2
2
2
2
2
2
2
2
2
2 £
2
2
2
2
2 2
2
2
2 y
1
1
2
2
2
0
0
0
1
1
1
2
2
2 a
1
2
0
1
2
0
1
2
0
1
2
0
1
2
GLF[U8], ;2 = 4d + 9, modulo 11. i* = ai + /?.
FIRST TABLE.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30 a
1
4
3
4
10
10
9
5
2
9
10
2
8
6
8
9
9
7
10
4
7
9
4
5
1
5
7
7 0
0
9
3
5
3
2
2
4
1
7
4
2
0
7
6
10 |
6
4
4 1 8 2
3
8 4
0
3
1
9
1
8 1 A
31
32
1 33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
1 60 a
3
9
8
3
7
8
10
2
10
3
3
6
7
5
6
3
5
9
4
9
6
6
1
3
10
1
6 J8
8
5
4
6
5
8
0
6
2
7
2
5
5
10
8
1
10
5
0
1
4
3
4
10
10
9 5
2
9
10 | A
61
62
63
64
65
66
1 67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
8Q
87
88
89
90 a
10
7
8
7
1
1
2
6
9
2
1
9
3
5
3
2
2
4
1
7
4
2
7
6
10
6
4
4 P
0
2
8
6
8
9
9
7
10
4
7
9
0
4
5
1
5
7
7 1
3
9
8
3
7 0
8
10
2
10
3 1 A
91
92
93
94
95
96
97
98
99
100
101
102
103
| 104
1 105
106
107
108
109
110
111
112
113
114
115
116
117
1 118
119
120 a
8
2
3
8
4
3
1
9
1
8
8
5
4
6
5
8
6
2
7
2
5
5
10
8
1
10
5 0
3
6
7
5
6
3
0
5
9
4
9
6
6
1
3
10
1
6
0
10
7
8
7
1
1
2
6
9
2
1
1905.] GALOIS FIELD TABLES. 31
GF[lP]y i2 = 4t + 9, modulo 11. ix = ai + P.
SECOND TABLE.
A
120
12
96
24
48
108
84
36
72
60
1
27
58
80
100
98
117
71
65
66
55
13
9
39
83
70
77
92
78
112 a
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2 18
1
2
3 4
5
Ö 7
8 9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8 I A
1 67
110
97
76
41
3 74
42
34
93
31
56
47
25
104
21
90
51
4
95
79
82
2
89
49
114
119
26
8
1 75 a
2
2
3
3
3
8
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5 0
9
10
0
1
2
3
4
5 6
7 8
9 10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5 A
103
i 113
45
28
106
109
46
88
105
53
43
15
68
86
59
54
85
29
62
22
19
35
64
111
30
81
44
37
107
1 116 a
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
7
7
8
8
8 0 1
6
7
8
9 10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2 A
91
33
94
102
14
63
101
16
73
50
7
52
18
32
17
10
23
99
69
61
115
6
5
11 57
38
40
20
118
1 87 a
8
8
8
8
8
8
8
8
9
9
9
9
9 9
9
9
9
9
9
10
10
10
10
10
10
10
10
10
10
10 Ê
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
GJFT58], i3 s 2i + 3, modulo 5. iA = «i2 + /?i + 7-
FIRST TABLE.
A
1
2
3
4
5
6
7
8
9
10
11
12 a
1
2
3
4
2
2
1
3
3 0
1
0
2
3
4
2
2
1
0
3
3
1 y
0
0
3
0 1
4 1 2
1
1
3
0
4 1 A
13 14
15
16
17
18
19
20
21
22
23
1 24 a
1
1
3
2
4
3
4
3
2
3
3 0
0
1
3
2
4
3
4
3
2
3
3
2 4
3
0
3
4
1
2
4
2
4
1
4 1 A
25
26
27
28
29
30
31
32
33
34
35
1 36 a
2
3
1
1
1
3
1
4 0
0
3
1
1
1
0
3
0
1
4
2 y
4
1
0
4
3
3
3
0
0
4
0
3
32 GALOIS FIELD TABLES. [Oct.,
GF[53], i3 = 2* + 3, modulo 5. tA = ot* + pi + y.
FIRST TABLE.—Continued.
A
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66 a
2
1
1
3
4
4
3
3
4
1
2
4
2
4
1
4
4
1
4
3
3
3
4
3 1
1
3
0
4
4
3
0
3
4
1
2
4
2
4
1
4
4
1
0
4
3
3
3
0
4
0
3
2 y
2 1 1
3
3
4
0
2
2
4
0
4
2
3
1
2
1
2
3
2 1
2
3
0
2
4
4
4
0
0
2
0 A
67
68
69
70
71
72
1 73 74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
1 96 a
2
1
3
3
4
2
2
4
4
2
3
1
2
1
2
3
2
2
3
2
4
4
4
2 0
1
3
3
4
0
2
2
4
0
4
2
3
1
2
1
2
3
2
2
3
0
2
4
4
4
0
2
0
4 V
4
1
3
4
4
2
0
1
1
2 0
2
1
4
3
1
3
1
4
1
1
4
0
1
2
2
2
0
0
1 A
97
98
99
100
101
102
103
104
i 105
i 106
107
108
109
110
111
112
113
114
115
116
f 117
118
119
120
121
122
123
124 a
4
1
3
4
4
2
1
1
2
2
1
4
3
1
3
1
4
1
1
4
1
2
2
2 J8
1
3
4
4
2
0
1
1
2
0
2
1
4
3
1
3
1
4
1
1
4
0
1
2
2
2
0 Y
0
2
3
4
2
2
1
0
3
3
1
0
1
3
2
4
3
4
3
2
3
3
2
0
3
1
1
1
SECOND TABLE.
A
124
93
31
62
1
103
119
14
34
94
107
72
3
88
32
26
65
10 a 0
1
1
1
1
1
2
2
2
2
2
3
3
3
3 y 1
1
2
3
4
0
1
2
3
4
0 1 1
2
3
4
0
1
2
3 A
45
63
1 96
76
57
41
2
1 9
56
30
13
104
38
116
29
28
120
1 82 a
1
1
1
1
1
1
1
1
1
1
1
1 0
3
4
4
4
4
4
0
0
0
0
0
1
1
1
1
1
2
2 y
4
0
1
2
3
4 0
1 1
2
3
4
0
1
2
3
4
0
1 A
48
i 105
1 80
15
68
98
39
112
35
109
53
117
114
95
123
102
106
1 25 a
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2 0
2
2
2
3
3
3
3
3
4
4
4
4
4
0
0
0
0
0 y
2
3
4
0
1
2
3
4
0
1
'2
3
4
0
1
2
3
4
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36 GALOIS FIELD TABLES. [Oct.,
GF[V], i7 = i + 1, modulo 2. ix = ai* + /3i5 + yi4 + ôtf + e# + Ci + *?.
SECOND TABLE.—Continued.
A
125
88
61
110
26
36
106
93
52
75
41
72
85
80
102
60 a
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 fi
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1 5
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0 e
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0 i
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0 y i
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0 A
124
105
25
40
51
101
84
24
123
83
50
49
122
120
121 a
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 fi
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 y
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 s
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1 e
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1 c
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1 V
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
GF[W], i' = i + 11, modulo 13. iK = ai + /?.
FIRST TABLE.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24 a
1
1
12
10
12
5
7
10
9
2
10
6
12
2
2
11
7
11
10
1
7
5
1 4 fi
0
11
11
2
G 2 1 3 I
12
6 8
9
6
1
2
0
9
9
4
12
4
6
11
12
1 3 25
26
27
28
29
30
1 31 32
33
1 34 I 35
36
37
38
39
1 40
1 4l 1 42 1 43 44
45
46
47
1 48 a
7
12
11
4
4
9
1
9
7
2
1
10
8
1
11
9
8
8
5
2
5
1 1 fi
5
12 1
2
4
0
5 1
ö 1 8
11
8
12
9
11
6 10
11
4
8 1 0
10
10
3
9
1 3 A
49
50
51
52
53
54
55
56
57 .
58
59
60
61
1 62
63
64
65
66
67
68
69
70
71
1 72 a
4
2
7
3
2
9
5
3
3
10
4
10
2
8
4
1
6
4
5
10
6
1 6 fi
11
5
9
12
7
9 1 8 I
3
0
7
7 6
5
6
9
10
5
11
1
5
3
6
0
1 1 A
73
74
75
76
77
78
79
80
81
82
83
84
i 85
| 86
87
88
89
90
91
92
93
94
95
1 96 a
7
8
7
4
3
8
2
12
8
10
7
12
12
1
3
1
8
6
3
4
11
3
7 fi
1
12
10
12
5
7
10
9
2
10
6
12
0
2
2
11
7
11
10
1
7
5
4
7
1905.] GALOIS FIELD TABLES. 37
OF [132], i2 = i + 11, modulo 13. iK = ai + /?.
FIRST TABLE.—Continued.
A
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114 a
1
11
11
2
6
2
3
12
6
8
9
6
1
2
9
9 0
12
11
o 4
4
9
1
9
7
2
1
10
8
1
11
9
0
8 1 A
115
116
117
118
119
120
121
122
123
124
125
126
127
128
1 129
130
131
| 132 a
4
12
4
6
11
12
3
5
12
2
4
5
5
8
11
8
12 £ '
8
5
2
5
1
4
2
7
3
2
9
5
0
3
3
10
4
10 1 A
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
| 150 a
9
11
6
10
11
4
8
10
10
3
9
3
11
5
9
12
7 0
2
8
4
1
6
4
5
10
0
6
6
7
8
7 4 '
3
8
2 I A
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
| 168 a
9
8
3
7
7
6
5
6
9
10
5
11
1
5
3
6 0
12
8
10
7
0
12
12
1
3
1
8
6
3
4
11
3
7
1
SECOND TABLE.
A
168
14
56
28
126
70
154
42
112
140
98
84
1
110
87
48
164
65
21
89
32
36
39
2
97 a
1
1
1
1
1
1
1
1
1
1
1
1
1 0
1
2
3
4
5
6
7
8
9
10
11
12
0
1
2
3
4 5
6
7 8
9
10
11
12 | 1 A
15
103
124
46
101
50
62
53
10
16
79
111
35
57
92
121
166
95
77
143
58
145
104
153
1 88 a
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3 00.
0
1
2
3
4
5
6
7
8
9
10
11
12
0
1
2
3
4
5
6
7 I 8
9
10
11 1 1 À
52
29
67
117
24
138
30
60
93
115
125
64
49
76
127
158
6
128
147
68
162
122
55
47
1 45 a
3
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5 0
12
0
1
2
3
4
5
6
7
8
9
10
H 12
0
1
2
3
4
5
6
7 8
9
10 1 A
165
23
71
72
106
159
135
118
12
167
109
102
91
66
157
155
73
150
7
18
25
83
96
34
1 51 a
5
5
6
6
6
6
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
7 00.
11
12
0
1
2
3
4
5
6
7
8
9
10
11
12
0
1
2
3
4
5
6
7
8
9
38 NOTES. [Oct.,
GF[W], i2 = i-\- 11, modulo 13. iK = ai + /?.
SECOND TABLE.—Continued.
A
75
22
156
43
107
81
129
131
139
38
78
152
63
44
90
74
113 a
7
7
7 8
8
8
8
8
8
8
8
8
8
8
8
8
9 J8
10
11
12
0
1
2
3
4
5
6
7 8
9 10
11
12
o 1 A
160
133
148
41
31
9 144
114
54
108
33
151
141
136
4
69
1 20 a
9
9
9
9
9
9
9
9
9
9
9
9
10
10
10
10
10 P
1
2
3
4
5
6
7
8
9
10
11
12
0
1
2
3
4 1 A
61
142
59
161
11
82
37
8
99
119
27
163
100
94
137
146
1 134 a
10
10
10
10
10
10
10
10
11
11
11
11
11
11
11
11
11 0
5
6
7
8
9
10
11
12
0
1
2
3
4
5
6
7
8 1 A
17
130
40
19
85
13
86
123
120
116
5
105
149
80
132
3
[ 26 a
11
11
11
11
12
12
12
12
12
12
12
12
12
12
12
12
12 J8
9
10
11
12
0
1
2
3
4
5
6
7
8
9
10
11
12
NOTES.
THE July number (volume 6, number 3) of the Transactions
of the AMERICAN MATHEMATICAL SOCIETY contains the fol
lowing papers : " Sur les lignes géodésiques des surfaces con
vexes/' by H. POINCARÉ ; " The classification of quadrics/' by
T. J. FA. BROMWICH ; " On differential invariants/' by J. E.
WRIOHT ; " Groups of order pm, which contain cyclic subgroups
of order ^>m-3/' by L. I. NEIKIRK ; "On the invariant sub
groups of prime index," by G. A. MILLER ; " On a general
method for treating transmitted motions and its application to
indirect perturbations/' by E. W. BROWN ; " On hypercomplex
number systems/' by L. E. DICKSON ; " A theorem on finite
algebras/' by J. H. MACLAGAN-WEDDERBURN ; " The relation
of the principles of logic to the foundations of geometry/' by
J. EOYCE ; " On multiple integrals/' by J. PIERPONT.
THE July number (volume 27, number 3) of the American
Journal of Mathematics contains : " Deduction of the power
series representing a function from special values of the latter/'
by G. W. HILL ; " On the definition of reducible hypercomplex
number systems/' by S. EPSTEEN and EL B. LEONARD;