Geometric Structures: Practice 8: Extensions of Finite Fields
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451
1. Write down the multiplication table and addition table for the integers modulo 3, otherwise known as Z3. Use the symbols 0, 1, and 2 for the elements of Z3. (Hint: Each table should have 9 entries)
2. Write down all the quadratic polynomials whose coefficients are in Z3. For each of these polynomials check which if any of the numbers 0, 1, and 2 are roots of the polynomial.
3. Verify that the polynomial x2 + x + 2 = 0 has no roots in Z3. Hence, we can create a new finite number system (finite field) with 9 elements where each element has the form a + b Δ where Δ2 = - Δ - 2 = 2Δ + 1, and a and b are elements of Z3. Some examples of these 9 numbers are: 0, 2, 1 + Δ, and 1 + 2Δ. This finite arithmetic is called GF (32 ).
Construct an addition and multiplication table for these 9 numbers. (Hint: These tables will each have 81 entries.)
Comment: One can use the finite field that you constructed above to construct a finite affine geometry (where there are unique parallels) with exactly 81 points and 90 lines. Each line will contain exactly 9 points. Ten lines will go through every point The points will have the form (x, y ) where x and y are elements of GF (32 ), and the lines will have the form ax + by + c = 0 where a, b, and c are not all 0 and are members of GF (32 ). One can then construct a finite projective plane with 91 points and 91 lines in the usual way by introducing homogeneous coordinates.