Extensions of Finite Fields

prepared by:

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451

Email: malkevitch@york.cuny.edu (for additions, suggestions, and corrections)

web page: www.york.cuny.edu/~malk

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 (unique parallels) geometry 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.