Geometric Structures: Finite Arithmetics
York College (CUNY)
Jamaica, New York 11451
There are many points of view for discussing numbers. These include the use of numbers in counting and measuring. Another important point of view is the role numbers play in thinking about the solutions of equations.
If one is given the equation x + 8 = 2, then this equation has no solution if x is confined to be a counting number number. However, x + 3 = 45 does have a solution among the counting numbers, namely x = 42. If an equation does not have a solution for a particular collection of numbers Y, one can ask whether or not one can extend Y to a new collection of numbers Y', which in some sense will contain Y, and where the equation will have a solution.
For the equation y + 8 = 2, one can extend the counting numbers to a bigger collection of numbers, usually denoted Z and called the integers where the equation does have a solution. Furthermore, there is a natural way in which the integers contain the counting numbers as a subset.
Now consider the equations 4x = 12 and 4x = 11 where x is restricted to Z. The first equation has the solution x = 3 while the second equation has no solution in Z. Yet, once again, one can extend Z to a new number system where 4x =11 does have a solution. This number system is usually denoted by Q, the rational numbers. One perspective on rational numbers is that they have the form a/b where a and b are members of Z with the restriction that b can not be 0. In a very formal setting one thinks of Q as being a "new system" with its own addition and multiplication which has a subset of numbers which behave exactly the way the elements of Z behave. Although this approach is rather "dry" it is necessary for mathematical precision. Here, I take a more informal approach.
The rational numbers, too, are not rich enough to enable one to solve all the equations that one might want to. Thus, the simple equation x2 = 2 has no solution when restricted to Q. This result is quite astonishing and surprised Greek geometers when they realized that an isosceles right triangle with leg 1 had a hypothenuse whose length was not rational. The numbers that make it possible to solve polynomial equations with integer coefficients are known as the algebraic numbers. Yet, they are not rich enough to capture all the numbers that arise naturally in Euclidean geometry. A circle of radius 1 has a circumference of length 2π. It turns out that π is neither a rational number nor an algebraic number, though neither of these facts is easy to show. Numbers such as π, belong to an extension of the rational number system, known as the real numbers. These numbers can be thought of as the "limits" of infinite sequences of rational numbers. In fact, there are a variety of approaches to constructing the real numbers. Those numbers which are real but are not algebraic are known as transcendental numbers. In an astonishing sense, made precise by Georg Cantor, there are many "more" transcendental numbers than algebraic real numbers.
Although the real numbers are a very rich collection of numbers with many nifty properties, they still do not enable one to solve some very simple equations. For example, x2 + 1 = 0 has no real number solution. The way to deal with this problem is again to extend an existing number system, creating a new number system which "contains" the real numbers and for which x2 + 1 = 0 does have a solution. This system is known as the complex numbers. The complex numbers have the form a + bi where a and b are real numbers and i is a special symbol with the property that i2 = -1. To multiply two complex numbers, one multiplies them as if they were polynomials and replaces any i2 that appears with -1. We will see that this "trick" enables one to construct many other interesting number system in a similar way.
The real numbers obey a lot of nice "rules" such as commutativity of addition and multiplication and associativity of addition and multiplication. Addition and multiplication are linked by a distributive law: a(b + c) = ab + ac. The real numbers, in addition to their algebraic properties, such as the ones just listed, also are ordered. This means that for any pair of real numbers, it is possible to say that either they are in fact the same real number or one is "bigger" than the other. Though the complex numbers share the nice algebraic properties of the real numbers, they are not ordered. The traditional name for an algebraic system with two operations + and x which obey the algebraic operations shared by the rational, real, and complex numbers is a field. There are a variety of ways to give a formal definition of a field. For those familiar with group theory the additive structure of a field is a group which obeys the commutative law (i.e. an Abelian group) and the multiplication is also a commutative group on the non-zero elements. The operations addition and multiplication are connected by the "distributive law": a ( b + c ) = ab + ac.
An interesting question is whether or not there are number systems which have the algebraic properties such as the rational, numbers, the real numbers and complex numbers obey. In addition to the properties mentioned above, these systems have special numbers denoted 0 and 1, where the equations x + a = 0 has a solution (-a) and ax = 1 (a not zero) has the solution x = 1/a. Thus, every element in the number system has an additive inverse and every element other than 0 has a multiplicative inverse.
The necessary concepts to show that there are indeed finite arithmetics which have the same algebraic properties as the real numbers or rational numbers are surprisingly new. The first tools were developed by Legendre and Gauss. and Euler: the theory of congruences. Rather than do this in general we will consider a specific example. Consider the prime number 5. (The primes are those positive integers 2 or more which have only 1 and themselves as divisors.) When an integer n is divided by 5, whether or not it is positive or negative, the number can be written in the form n = 5q + r where q is an integer and r is an integers satisfying 0 ≤ r < 5. Thus, any integer when divided by 5 can be thought of as being associated with either 0, 1, 2, 3, or 4. The standard way to express this is using the idea of congruence, When two numbers a and b have the same remainder when divided by m, we write a congruent to b mod m. (The notation of using three parallel bars is due to Gauss.)
If one takes any integer it is congruent to 0, 1, 2, 3, or 4 modulo m. The next step is to treat the classes into which all the integers are partitioned by what remainder they have mod 5 as if they were numbers! These numbers will be denoted by the same digit in boldface: 0, 1, 2, 3, and 4. To find the sum or product of two of these numbers we merely take any integer that leaves this remainder mod 5 as the numbers being added or multiplied, perform the usual addition or multiplication on these, taking as the answer, the value obtained mod 5! Thus, to find the product of 2 and 4 we can take the product of 7 and 9 (which leave remainders of 2 and 4 when divided by 5). Since 7(9) is 63, and 63 divided by 5 leaves the remainder of 3, the product of 2 and 4 is 3. For this approach to be valid we have to prove that had we used, for example, any other integer than 7 whose remainder is 2 when divided by 5 to represent 2 that we would get the same answer. It is not difficult to provide these proofs. One easily can construct the 5x5 addition and multiplication facts tables for the "bold" numbers, which is known as Z5!
I mentioned above that for the reals and rational numbers there were additive inverses and multiplicative inverses for non-zero values. Is that true in Z5? It turns out the answer is yes. For example, since 3 + 2 = 0 we see that 2 is the additive inverse of 3 and 3 is the additive inverse of 2. Also, since 2 times 3 equals 1, it follows that 2 is the multiplicative inverse of 3 and 3 is the multiplicative inverse of 2. In fact, the numbers in Z5 obey all of the algebraic properties of the real numbers, the complex numbers, or the rational numbers, although, like the complex numbers, they can not be ordered.
It turns out that there is a different finite arithmetic for each prime p, and the arithmetic associated with the prime p is called Zp. Are there other finite arithmetics that also are algebraically like the rationals or reals? The surprising answer is yes. In fact, infinitely many additional such fields. The general result is that for each prime power, pk, there is a finite field. When k = 1 we have seen how to construct these. How can one get the others?
Here is a way to see how to construct a finite field of 4 elements. Start with the finite field of two elements, Z2. Now we look at the polynomial 1x2 + 1x + 1 = 0. This polynomial has no root in Z2. This can be checked by substitution of the two elements in the field into the polynomial and verifying that neither of these numbers satisfies it. So now, we proceed just the way we did to construct the complex numbers over the reals. For convenience we will no longer use bold numerals to indicate the numbers in Z2. Define λ 2= λ + 1. Recall that we have -1 = 1 in Z2. The elements of our new field will have the form a + bλ where a and b are selected from Z2. Since a and b can only take on two values we have exactly 4 elements in our new arithmetic: 0, 1, λ, and 1+ λ. It is not difficult to check that these 4 numbers each has an additive inverse. In fact, adding each of the elements to itself results in the value 0. For multiplication in this 4 number arithmetic, to compute for example λ(1+ λ) we get λ + λ2 = λ + λ + 1 (as we defined it above) = 1. Thus, λ and λ + 1 are multiplicative inverses of each other. It is good practice to make up the 4x4 addition and multiplication tables for this 4 element arithmetic.
Recall that because we are working in Z2, 1 + 1 = 0 (remember strictly speaking we should be using bold face but have dropped it out of convenience).
A polynomial (degree at least 2) is called irreducible over a field (see above) if it has coefficients taken from the field but can not be factored into lower degree polynomials with coefficients from that field. In particular, the polynomial has no roots in the "base" field. It turns out that for every; finite arithmetic Zp there is an kth degree polynomial which is irreducible over Zp. Using this fact one can construct a finite arithmetic (finite field) with pk elements as explained above. These fields are traditionally known as the Galois Fields, honoring Erviste Galois who was a pioneer in their study.
Finite arithmetics find applications in the theory of error correcting codes, and also in the theory of finite geometries. For example, one can imitate the approach used in analytical geometry to construct the Euclidean plane by using for points, ordered pairs (x, y ) of real numbers and for lines equations of first degree: a x + by + c = 0 where not both a and b are 0. Thus, if one uses elements from the field Z5 and defines points to be ordered pairs (x, y) where the coordinates are taken from Z5 and lines have the form ax + by + c = 0 where a, b, and C are again taken from Z5 and a and b are not both 0. Since Z5 has 5 elements it is easy to check that one gets exactly 25 points and exactly 30 lines. The geometry one gets obeys the very nice rules that:
a. Given two point there is exactly one line which contains them.
b. Given a point P not on a line l, there is a unique line through P which is parallel to l.
One can use the concept of slope, in this finite geometry, to tell when lines are parallel in much the same way that one uses it in analytical geometry. For example, in the 25 point geometry mentioned above, if one wants the line through (2, 4 ) parallel to the line y = 1 x + 4 the answer would be y = 1 x + 2. Also, one can find exactly what 5 points lie on the line y = 1 x + 4. These points are ( 0, 4), ( 1, 0 ), ( 2, 1 ), ( 3, 2 ), ( 4, 3 ). Remember that just as in analytical geometry there is a collection of lines with undefined slope. Since for each of the 5 possible slopes there are 5 lines with that slope and there are 5 lines with undefined slope, there are exactly 30 lines in this plane.
One can take this 25 point plane and construct a 31 point plane by adding one new line and putting one additional point on each line of the old plane. This geometry, is a projective geometry, and has 31 points and 31 lines. One can use as coordinates for this plane, triples of elements from the field Z5 except for the triple of all 0's.
Gallian, J., Contemporary Abstract Algebra, Heath, Lexington.
Rosen, K., Elementary Number Theory and its Applications, Addison-Wesley, Reading.