Some Small Finite Fields
F2:
F3:
| + |
0 |
1 |
2 |
| 0 |
0 |
1 |
2 |
| 1 |
1 |
2 |
0 |
| 2 |
2 |
0 |
1 |
|
| × |
0 |
1 |
2 |
| 0 |
0 |
0 |
0 |
| 1 |
0 |
1 |
2 |
| 2 |
0 |
2 |
1 |
|
F4:
| + |
0 |
1 |
A |
B |
| 0 |
0 |
1 |
A |
B |
| 1 |
1 |
0 |
B |
A |
| A |
A |
B |
0 |
1 |
| B |
B |
A |
1 |
0 |
|
| × |
0 |
1 |
A |
B |
| 0 |
0 |
0 |
0 |
0 |
| 1 |
0 |
1 |
A |
B |
| A |
0 |
A |
B |
1 |
| B |
0 |
B |
1 |
A |
|
Field of 8 elements represented as matrices
integers are modulo 2
element (0) element (1) element (2) element (3) 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 1 1
element (4) element (5) element (6) element (7) 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 0 0 0 1 0
+/ (0) (1) (2) (3) (4) (5) (6) (7)
(0) 0 1 2 3 4 5 6 7
(1) 1 0 4 7 2 6 5 3
(2) 2 4 0 5 1 3 7 6
(3) 3 7 5 0 6 2 4 1
(4) 4 2 1 6 0 7 3 5
(5) 5 6 3 2 7 0 1 4
(6) 6 5 7 4 3 1 0 2
(7) 7 3 6 1 5 4 2 0
x/ (0) (1) (2) (3) (4) (5) (6) (7)
(0) 0 0 0 0 0 0 0 0
(1) 0 1 2 3 4 5 6 7
(2) 0 2 3 4 5 6 7 1
(3) 0 3 4 5 6 7 1 2
(4) 0 4 5 6 7 1 2 3
(5) 0 5 6 7 1 2 3 4
(6) 0 6 7 1 2 3 4 5
(7) 0 7 1 2 3 4 5 6
_________________________________________________________
Field of 9 elements represented as matrices
integers are modulo 3
element (0) element (1) element (2) 0 0 1 0 0 1 0 0 0 1 1 1
element (3) element (4) element (5) 1 1 1 2 2 0 1 2 2 0 0 2
element (6) element (7) element (8) 0 2 2 2 2 1 2 2 2 1 1 0
+/ (0) (1) (2) (3) (4) (5) (6) (7) (8)
(0) 0 1 2 3 4 5 6 7 8
(1) 1 5 3 8 7 0 4 6 2
(2) 2 3 6 4 1 8 0 5 7
(3) 3 8 4 7 5 2 1 0 6
(4) 4 7 1 5 8 6 3 2 0
(5) 5 0 8 2 6 1 7 4 3
(6) 6 4 0 1 3 7 2 8 5
(7) 7 6 5 0 2 4 8 3 1
(8) 8 2 7 6 0 3 5 1 4
x/ (0) (1) (2) (3) (4) (5) (6) (7) (8)
(0) 0 0 0 0 0 0 0 0 0
(1) 0 1 2 3 4 5 6 7 8
(2) 0 2 3 4 5 6 7 8 1
(3) 0 3 4 5 6 7 8 1 2
(4) 0 4 5 6 7 8 1 2 3
(5) 0 5 6 7 8 1 2 3 4
(6) 0 6 7 8 1 2 3 4 5
(7) 0 7 8 1 2 3 4 5 6
(8) 0 8 1 2 3 4 5 6 7
_________________________________________________________
F16 is represented by the polynomials a + b x + c x2 + d x3.
a, b, c, and d are integers modulo 2
The polynomials are generated by the powers of x using the rule
x4 = 1 + x.
e ( 0) e ( 1) e ( 2) e ( 3)
e ( 4) e ( 5) e ( 6) e ( 7)
e ( 8) e ( 9) e (10) e (11)
e (12) e (13) e (14) e (15)
+/ 0_ 1_ 2_ 3_ 4_ 5_ 6_ 7_ 8_ 9_10_11_12_13_14_15_ 0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1_ 1 0 5 9 15 2 11 14 10 3 8 6 13 12 7 4 2_ 2 5 0 6 10 1 3 12 15 11 4 9 7 14 13 8 3_ 3 9 6 0 7 11 2 4 13 1 12 5 10 8 15 14 4_ 4 15 10 7 0 8 12 3 5 14 2 13 6 11 9 1 5_ 5 2 1 11 8 0 9 13 4 6 15 3 14 7 12 10 6_ 6 11 3 2 12 9 0 10 14 5 7 1 4 15 8 13 7_ 7 14 12 4 3 13 10 0 11 15 6 8 2 5 1 9 8_ 8 10 15 13 5 4 14 11 0 12 1 7 9 3 6 2 9_ 9 3 11 1 14 6 5 15 12 0 13 2 8 10 4 7
10_ 10 8 4 12 2 15 7 6 1 13 0 14 3 9 11 5
11_ 11 6 9 5 13 3 1 8 7 2 14 0 15 4 10 12
12_ 12 13 7 10 6 14 4 2 9 8 3 15 0 1 5 11
13_ 13 12 14 8 11 7 15 5 3 10 9 4 1 0 2 6
14_ 14 7 13 15 9 12 8 1 6 4 11 10 5 2 0 3
15_ 15 4 8 14 1 10 13 9 2 7 5 12 11 6 3 0
x/ 0_ 1_ 2_ 3_ 4_ 5_ 6_ 7_ 8_ 9_10_11_12_13_14_15_ 0_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2_ 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 3_ 0 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 4_ 0 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 5_ 0 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 6_ 0 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 7_ 0 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 8_ 0 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 9_ 0 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8
10_ 0 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9
11_ 0 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10
12_ 0 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11
13_ 0 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12
14_ 0 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13
15_ 0 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14
_________________________________________________________
F25 represented by the numbers a + b√2, a and b are integers modulo 5
generated by powers of 2 + √2
| e ( 0) |
e ( 1) |
e ( 2) |
e ( 3) |
e ( 4) |
| 0 + 0√2 |
1 + 0√2 |
2 + 1√2 |
1 + 4√2 |
0 + 4√2 |
| e ( 5) |
e ( 6) |
e ( 7) |
e ( 8) |
e ( 9) |
| 3 + 3√2 |
2 + 4√2 |
2 + 0√2 |
4 + 2√2 |
2 + 3√2 |
| e (10) |
e (11) |
e (12) |
e (13) |
e (14) |
| 0 + 3√2 |
1 + 1√2 |
4 + 3√2 |
4 + 0√2 |
3 + 4√2 |
| e (15) |
e (16) |
e (17) |
e (18) |
e (19) |
| 4 + 1√2 |
0 + 1√2 |
2 + 2√2 |
3 + 1√2 |
3 + 0√2 |
| e (20) |
e (21) |
e (22) |
e (23) |
e (24) |
| 1 + 3√2 |
3 + 2√2 |
0 + 2√2 |
4 + 4√2 |
1 + 2√2 |
+/ 0_ 1_ 2_ 3_ 4_ 5_ 6_ 7_ 8_ 9_10_11_12_13_14_15_16_17_18_19_20_21_22_23_24_ 0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1_ 1 7 18 6 3 12 14 19 22 5 20 2 10 0 23 16 11 21 15 13 9 8 24 4 17 2_ 2 18 8 19 7 4 13 15 20 23 6 21 3 11 0 24 17 12 22 16 14 10 9 1 5 3_ 3 6 19 9 20 8 5 14 16 21 24 7 22 4 12 0 1 18 13 23 17 15 11 10 2 4_ 4 3 7 20 10 21 9 6 15 17 22 1 8 23 5 13 0 2 19 14 24 18 16 12 11 5_ 5 12 4 8 21 11 22 10 7 16 18 23 2 9 24 6 14 0 3 20 15 1 19 17 13 6_ 6 14 13 5 9 22 12 23 11 8 17 19 24 3 10 1 7 15 0 4 21 16 2 20 18 7_ 7 19 15 14 6 10 23 13 24 12 9 18 20 1 4 11 2 8 16 0 5 22 17 3 21 8_ 8 22 20 16 15 7 11 24 14 1 13 10 19 21 2 5 12 3 9 17 0 6 23 18 4 9_ 9 5 23 21 17 16 8 12 1 15 2 14 11 20 22 3 6 13 4 10 18 0 7 24 19
10_ 10 20 6 24 22 18 17 9 13 2 16 3 15 12 21 23 4 7 14 5 11 19 0 8 1
11_ 11 2 21 7 1 23 19 18 10 14 3 17 4 16 13 22 24 5 8 15 6 12 20 0 9
12_ 12 10 3 22 8 2 24 20 19 11 15 4 18 5 17 14 23 1 6 9 16 7 13 21 0
13_ 13 0 11 4 23 9 3 1 21 20 12 16 5 19 6 18 15 24 2 7 10 17 8 14 22
14_ 14 23 0 12 5 24 10 4 2 22 21 13 17 6 20 7 19 16 1 3 8 11 18 9 15
15_ 15 16 24 0 13 6 1 11 5 3 23 22 14 18 7 21 8 20 17 2 4 9 12 19 10
16_ 16 11 17 1 0 14 7 2 12 6 4 24 23 15 19 8 22 9 21 18 3 5 10 13 20
17_ 17 21 12 18 2 0 15 8 3 13 7 5 1 24 16 20 9 23 10 22 19 4 6 11 14
18_ 18 15 22 13 19 3 0 16 9 4 14 8 6 2 1 17 21 10 24 11 23 20 5 7 12
19_ 19 13 16 23 14 20 4 0 17 10 5 15 9 7 3 2 18 22 11 1 12 24 21 6 8
20_ 20 9 14 17 24 15 21 5 0 18 11 6 16 10 8 4 3 19 23 12 2 13 1 22 7
21_ 21 8 10 15 18 1 16 22 6 0 19 12 7 17 11 9 5 4 20 24 13 3 14 2 23
22_ 22 24 9 11 16 19 2 17 23 7 0 20 13 8 18 12 10 6 5 21 1 14 4 15 3
23_ 23 4 1 10 12 17 20 3 18 24 8 0 21 14 9 19 13 11 7 6 22 2 15 5 16
24_ 24 17 5 2 11 13 18 21 4 19 1 9 0 22 15 10 20 14 12 8 7 23 3 16 6
x/ 0_ 1_ 2_ 3_ 4_ 5_ 6_ 7_ 8_ 9_10_11_12_13_14_15_16_17_18_19_20_21_22_23_24_ 0_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2_ 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 3_ 0 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 4_ 0 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 5_ 0 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 6_ 0 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 7_ 0 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 8_ 0 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 9_ 0 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8
10_ 0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9
11_ 0 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10
12_ 0 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11
13_ 0 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12
14_ 0 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13
15_ 0 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14
16_ 0 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
17_ 0 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
18_ 0 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
19_ 0 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
20_ 0 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
21_ 0 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
22_ 0 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
23_ 0 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
24_ 0 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23