On the 27-element set {0,1,2,...,25,26}, use the following unusual definitions of "multiplication" and "addition":
Multiplication table:
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 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 25 26 2: 0 2 1 6 8 7 3 5 4 18 20 19 24 26 25 21 23 22 9 11 10 15 17 16 12 14 13 3: 0 3 6 9 12 15 18 21 24 11 14 17 20 23 26 2 5 8 19 22 25 1 4 7 10 13 16 4: 0 4 8 12 16 11 24 19 23 20 21 25 5 6 1 17 9 13 10 14 15 22 26 18 7 2 3 5: 0 5 7 15 11 13 21 26 19 2 4 6 17 10 12 23 25 18 1 3 8 16 9 14 22 24 20 6: 0 6 3 18 24 21 9 15 12 19 25 22 10 16 13 1 7 4 11 17 14 2 8 5 20 26 23 7: 0 7 5 21 19 26 15 13 11 1 8 3 22 20 24 16 14 9 2 6 4 23 18 25 17 12 10 8: 0 8 4 24 23 19 12 11 16 10 15 14 7 3 2 22 18 26 20 25 21 17 13 9 5 1 6 9: 0 9 18 11 20 2 19 1 10 17 26 8 25 7 16 6 15 24 22 4 13 3 12 21 14 23 5 10: 0 10 20 14 21 4 25 8 15 26 6 16 1 11 18 12 22 5 13 23 3 24 7 17 2 9 19 11: 0 11 19 17 25 6 22 3 14 8 16 24 13 21 5 18 2 10 4 12 23 9 20 1 26 7 15 12: 0 12 24 20 5 17 10 22 7 25 1 13 15 18 3 8 11 23 14 26 2 4 16 19 21 6 9 13: 0 13 26 23 6 10 16 20 3 7 11 21 18 4 17 14 24 1 5 15 19 25 2 12 9 22 8 14: 0 14 25 26 1 12 13 24 2 16 18 5 3 17 19 20 4 15 23 7 9 10 21 8 6 11 22 15: 0 15 21 2 17 23 1 16 22 6 12 18 8 14 20 7 13 19 3 9 24 5 11 26 4 10 25 16: 0 16 23 5 9 25 7 14 18 15 22 2 11 24 4 13 20 6 21 1 17 26 3 10 19 8 12 17: 0 17 22 8 13 18 4 9 26 24 5 10 23 1 15 19 6 14 12 20 7 11 25 3 16 21 2 18: 0 18 9 19 10 1 11 2 20 22 13 4 14 5 23 3 21 12 17 8 26 6 24 15 25 16 7 19: 0 19 11 22 14 3 17 6 25 4 23 12 26 15 7 9 1 20 8 24 16 18 10 2 13 5 21 20: 0 20 10 25 15 8 14 4 21 13 3 23 2 19 9 24 17 7 26 16 6 12 5 22 1 18 11 21: 0 21 15 1 22 16 2 23 17 3 24 9 4 25 10 5 26 11 6 18 12 7 19 13 8 20 14 22: 0 22 17 4 26 9 8 18 13 12 7 20 16 2 21 11 3 25 24 10 5 19 14 6 23 15 1 23: 0 23 16 7 18 14 5 25 9 21 17 1 19 12 8 26 10 3 15 2 22 13 6 20 11 4 24 24: 0 24 12 10 7 22 20 17 5 14 2 26 21 9 6 4 19 16 25 13 1 8 23 11 15 3 18 25: 0 25 14 13 2 24 26 12 1 23 9 7 6 22 11 10 8 21 16 5 18 20 15 4 3 19 17 26: 0 26 13 16 3 20 23 10 6 5 19 15 9 8 22 25 12 2 7 21 11 14 1 24 18 17 4Addition table:
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 25 26 1: 1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24 2: 2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25 3: 3 4 5 6 7 8 0 1 2 12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20 4: 4 5 3 7 8 6 1 2 0 13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18 5: 5 3 4 8 6 7 2 0 1 14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19 6: 6 7 8 0 1 2 3 4 5 15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23 7: 7 8 6 1 2 0 4 5 3 16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21 8: 8 6 7 2 0 1 5 3 4 17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22 9: 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 0 1 2 3 4 5 6 7 8 10: 10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24 1 2 0 4 5 3 7 8 6 11: 11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25 2 0 1 5 3 4 8 6 7 12: 12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20 3 4 5 6 7 8 0 1 2 13: 13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18 4 5 3 7 8 6 1 2 0 14: 14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19 5 3 4 8 6 7 2 0 1 15: 15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23 6 7 8 0 1 2 3 4 5 16: 16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21 7 8 6 1 2 0 4 5 3 17: 17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22 8 6 7 2 0 1 5 3 4 18: 18 19 20 21 22 23 24 25 26 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19: 19 20 18 22 23 21 25 26 24 1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15 20: 20 18 19 23 21 22 26 24 25 2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16 21: 21 22 23 24 25 26 18 19 20 3 4 5 6 7 8 0 1 2 12 13 14 15 16 17 9 10 11 22: 22 23 21 25 26 24 19 20 18 4 5 3 7 8 6 1 2 0 13 14 12 16 17 15 10 11 9 23: 23 21 22 26 24 25 20 18 19 5 3 4 8 6 7 2 0 1 14 12 13 17 15 16 11 9 10 24: 24 25 26 18 19 20 21 22 23 6 7 8 0 1 2 3 4 5 15 16 17 9 10 11 12 13 14 25: 25 26 24 19 20 18 22 23 21 7 8 6 1 2 0 4 5 3 16 17 15 10 11 9 13 14 12 26: 26 24 25 20 18 19 23 21 22 8 6 7 2 0 1 5 3 4 17 15 16 11 9 10 14 12 13
You may now verify that, as usual, 1x=x1=x, 0x=x0=0, 0+x=x+0=x, xy=yx, (xy)z=x(yz), x+y=y+x, (x+y)+z=x+(y+z), and x(y+z)=xy+xz, and that each row (or column) of the + table (and multiplication table) contains every field element exactly once (except for the 0-row and 0-column of the mul table). I.e. this really is a "field."
The squares are 0, 1, 4, 6, 7, 9, 13, 14, 15, 16, 17, 19, 20, and 24.
A famous theorem of Galois states that the multiplicative group of a finite field always is cyclic, a fact you can verify thusly in our case: 326=30=1, 31=3, 32=9, 33=11, 34=17, 35=8, 36=24, 37=10, 38=14, 39=26, 310=16, 311=5, 312=15, 313=2, 314=6, 315=18, 316=19, 317=22, 318=4, 319=12, 320=20, 321=25, 322=13, 323=23, 324=7, 325=21.