Number theory has already been seen in earlier chapters of this book. It was used in the design of fast Fourier transform algorithms. We did make use of some ideas that only now will be proved. This chapter, which is a mathematical interlude, will develop the basic facts of number theory – some that were used earlier in the book and some that we may need later.

We also return to the study of fields to develop the topic of an extension field more fully. The structure of algebraic fields will be important to the construction of number theory transforms in Chapter 10 and also to the construction of some multidimensional convolution algorithms in Chapter 11 and for some multidimensional Fourier transform algorithms in Chapter 12.

Elementary number theory

Within the integer quotient ring Zq, some of the elements may be coprime to q, and, unless q is a prime, others will divide q. It is important to us to know how many elements there are of each type.

Definition 9.1.1 (Euler)The totient function, denoted ϕ(q), where q is an integer larger than one, is the number of nonzero elements in Zq that are coprime to q. For q equal to one, ϕ(q) = 1.

When q is a prime p, then all the nonzero elements of Zq are coprime to p, and so ϕ(p) = p – 1 whenever p is a prime.