4. Fields

In the natural numbers we can add and multiply, but not always subtract or divide. If we extend these to the integers, then we can also always subtract. But, for the rational numbers (all fractions), we can freely use all four arithmetical operations (with the exception of dividing by zero). If we turn back to Galois, then it was in connection with solving equations that it was necessary to look at systems of numbers, greater than the rational numbers, but less than the real numbers, where one can freely add, subtract, multiply and divide. Such a system is called a number field, or just a field. Systems where one can do all four of these operations are exactly what it takes to make the mathematics work in many contexts. For example, everything to do with solving linear systems of equations works over a field.

And lo and behold, a new world appears here as well. If one takes a prime number p and looks at the numbers 0, ..., p-1, then adding or multiplying two such numbers and taking the remainder after division by p gives one of the numbers among 0, ..., p-1 again. In this system, one can also freely subtract and, it turns out, divide (except by zero). This gives us a basic example of a finite field of the type that ones studies and classifies in the course MAT220 Algebra. Finite fields, and vector spaces over these, are the fundamental algebraic structures in coding theory.

Read on about matrices.