Geometric Structures: Finite Fields
prepared by:
Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451
email:
malkevitch@york.cuny.edu
web page:
http://york.cuny.edu/~malk/
The rational numbers and the real numbers have lots of nice of algebraic properties. They are examples of an algebraic structure called a field. Since one can multiply and divide in a field on can solve equations such as 3x - 4 = 11, or 3x = 2. Are there finite algebraic structures which are fields, and which would allow for the solution of linear equations?
It turns out that there are finite fields. Today they are known as Galois Fields, for Erviste Galois who pioneered their study. One infinite class of finite fields are relatively easy to describe. If one looks at the equivalence classes of the integers modulo a prime p then these classes can be thought of as elements of a finite field.
To illustrate this, consider the integers modulo 3, often denoted Z_{3}. We will use the symbols 0, 1,and 2 to represent the sets of integers which leave remainders 0, 1, and 2 respectively, when divided by 3. We can define an addition and a multiplication on these classes. To multiply, for example, 2 and 2, we can take any elements of these classes, compute the remainder mod 3, and call this the "product." Thus, since 2 x 2 = 4 which has remainder 1 modulo 3, we conclude that the produce of 2 and 2 is 1. (We could have chosen different representatives from the same class, say 8 and 11 (each leaves a remainder of 2 when divided by 3) and the product is 88 which leaves a remainder of 1 when divided by 3.)
In a similar manner we can construct the addition and multiplication tables below:
+ | 0 | 1 | 2 |
0 | 0 | 1 | 2 |
1 | 1 | 2 | 0 |
2 | 2 | 0 | 1 |
(Note that since 1 + 2 = 0 we can thank of 1 and 2 as additive inverses. This means we can think of 1 as the negative of 2 (i.e. 1=-2) and 2 as the negative of 1 (i.e. 2 = -1).
x | 0 | 1 | 2 |
0 | 0 | 0 | 0 |
1 | 0 | 1 | 2 |
2 | 0 | 2 | 1 |
(Note that since 2 x 2 = 1, we can think of 2 as the multiplicative inverse of 2, and, thus. the "fraction 1/2" is 2 ins this system.)
Are there any other finite fields. It turns out the answer is yes. There are finite fields for any power of any prime. The idea behind seeing this is an extension of the reasoning that is used to construct the complex numbers from the real numbers. In this case, since x^{2} + 1 = 0 does not factor over the reals, we can define a new symbol i, such that i^{2} = -1, and the complex numbers take the form: a + bi, where a and b are real numbers. We treat the numbers a + bi as if they were "polynomials" and replace powers of i, using i^{2} = -1.
We can use the same approach to construct an "extension field" of a field Z_{p}. This is done by finding a polynomial with coefficients in Z_{p} which does not factor as two polynomials with coefficients from Z_{p}. Such a polynomial is said to be irreducible. To illustrate this process consider the polynomial x^{2} + 1 = 0. It is easy to check that with regard to Z_{3} neither of 1 nor 2 are roots of this polynomial. (1^{2} + 1 is not 0 nor is 2^{2} + 1). Hence, this is an irreducible polynomial with respect to Z_{3}. Thus, let us invent the symbol m, so that m^{2} = "-1" = 2. The numbers we get are now of the form a + bm where a and b are in Z_{3}. Since a and b can each take on any of three values, we get 9 different numbers. Thus, it turns out we will have a field with 9 elements
Here are some typical elements of this field:
2 + 2m; 1 + m; 2 + 0m.
What is the sum of, say, 2 + 2m and 2? The answer is 1 + 2m since 2 + 2 = 1.
What is the product of 2 + 2m and 1 + m? We get, treating the two as polynomials: 2 + 2m +2m + 2m^{2} = 2 + m + 2(2) = 2 + 1 + m = m.
If one works out the whole multiplication table and addition table you can verify that the properties of a field hold!
Using more abstract methods one can verify the general result that this example suggests: for every prime (p) power, one can construct a finite field. The method consists of finding a polynomial with coefficients in Z_{p}, which is irreducible and proceeding in the manner above.
There are many applications of finite fields. Among the most important of these is in the construction of error correcting codes, and in coordinatizing the points of finite projective planes and finite affine planes. In fact, using finite fields, it is possible to construct finite affine planes with n points on every line where n is a prime power, and use the usual addition of a line at infinity construction to construct finite projective planes with a prime power plus one points on every line. It turns out that these projective planes all obey Desargues and Pappus' Theorems (except for the cases where there are only a small number of points on a line).