Z can describe data structures. In this chapter we'll use Z to define the fundamental objects of computer graphics and computational geometry: points, line segments, contours, and polygons.

Consider the distinction between a contour, which is any sequence of connected line segments, and a polygon, which is a closed contour that has an inside and an outside (Figure 19.1). Your eye can see the difference immediately, and the distinction is vital for many computations of great practical importance.

Figure 19.2 is a computer graphic that shows a view of a patient's anatomy and radiation beam geometry, used to plan this patient's radiation treatment for cancer. Cross-sections of anatomical structures must be polygons, not just contours, and the physics calculations that compute the radiation dose depend on this. If some contours are not closed, or cross over themselves, the dose calculations may be incorrect.

The difference between contours and polygons is vital, but there is no way to express this distinction in most programming languages: You have to represent both as mere sequences (arrays or lists) of points. Data types in programming languages correspond closely to the way data are represented in computer memory: If two objects are stored in the same format, they belong to the same data type. Z is far more expressive because we can distinguish data types based on their values and constraints between the values of their components. This enables us to define data types that capture such requirements as “a closed contour that doesn't cross over itself.”

This chapter is of central importance since it contains various fundamental properties of finite fields and a description of methods for constructing finite fields.

The field of integers modulo a prime number is, of course, the most familiar example of a finite field, but many of its properties extend to arbitrary finite fields. The characterization of finite fields (see Section 1) shows that every finite field is of prime-power order and that, conversely, for every prime power there exists a finite field whose number of elements is exactly that prime power. Furthermore, finite fields with the same number of elements are isomorphic and may therefore be identified. The next two sections provide information on roots of irreducible polynomials, leading to an interpretation of finite fields as splitting fields of irreducible polynomials, and on traces, norms, and bases relative to field extensions.

Section 4 treats roots of unity from the viewpoint of general field theory, which will be needed occasionally in Section 6 as well as in Chapter 5. Section 5 presents different ways of representing the elements of a finite field. In Section 6 we give two proofs of the famous theorem of Wedderburn according to which every finite division ring is a field.

Many discussions in this chapter will be followed up, continued, and partly generalized in later chapters.

CHARACTERIZATION OF FINITE FIELDS

In the previous chapter we have already encountered a basic class of finite fields—that is, of fields with finitely many elements.

Most modern algebra texts devote a few pages (but no more) to finite fields. So at first it may come as a surprise to see an entire book on the subject, and even more for it to appear in the Encyclopedia of Mathematics and Its Applications. But the reader of this book will find that the authors performed the very timely task of drawing together the different threads of development that have emanated from the subject. Foremost among these developments is the rapid growth of coding theory which already has been treated in R. J. McEliece's volume in this series. The present volume deals with coding theory in the wider context of polynomial theory over finite fields, and also establishes the connection with linear recurring series and shift registers.

On the pure side there is a good deal of number theory that is most naturally expressed in terms of finite fields. Much of this—for example, equations over finite fields and exponential sums—can serve as a paradigm for the more general case; and the authors have gone as far in their treatment as is reasonable, using elementary algebraic methods only. As a result the book can also serve as an introduction to these topics.

But finite fields also have properties that are not shared with other types of algebra; thus they (like finite Boolean algebras) are functionally complete. This means that every mapping of a finite field can be expressed as a polynomial.

This introductory chapter contains a survey of some basic algebraic concepts that will be employed throughout the book. Elementary algebra uses the operations of arithmetic such as addition and multiplication, but replaces particular numbers by symbols and thereby obtains formulas that, by substitution, provide solutions to specific numerical problems. In modern algebra the level of abstraction is raised further: instead of dealing with the familiar operations on real numbers, one treats general operations —processes of combining two or more elements to yield another element—in general sets. The aim is to study the common properties of all systems consisting of sets on which are defined a fixed number of operations interrelated in some definite way—for instance, sets with two binary operations behaving like + and · for the real numbers.

Only the most fundamental definitions and properties of algebraic systems—that is, of sets together with one or more operations on the set—will be introduced, and the theory will be discussed only to the extent needed for our special purposes in the study of finite fields later on. We state some standard results without proof. With regard to sets we adopt the naive standpoint. We use the following sets of numbers: the set ℕ of natural numbers, the set ℤ of integers, the set ℚ of rational numbers, the set ℝ of real numbers, and the set ℂ of complex numbers.

GROUPS

In the set of all integers the two operations addition and multiplication are well known.

Any nonconstant polynomial over a field can be expressed as a product of irreducible polynomials. In the case of finite fields, some reasonably efficient algorithms can be devised for the actual calculation of the irreducible factors of a given polynomial of positive degree.

The availability of feasible factorization algorithms for polynomials over finite fields is important for coding theory and for the study of linear recurrence relations in finite fields. Beyond the realm of finite fields, there are various computational problems in algebra and number theory that depend in one way or another on the factorization of polynomials over finite fields. We mention the factorization of polynomials over the ring of integers, the determination of the decomposition of rational primes in algebraic number fields, the calculation of the Galois group of an equation over the rationals, and the construction of field extensions.

We shall present several algorithms for the factorization of polynomials over finite fields. The decision on the choice of algorithm for a specific factorization problem usually depends on whether the underlying finite field is “small” or “large.” In Section 1 we describe those algorithms that are better adapted to “small” finite fields and in the next section those that work better for “large” finite fields. Some of these algorithms reduce the problem of factoring polynomials to that of finding the roots of certain other polynomials. Therefore, Section 3 is devoted to the discussion of the latter problem from the computational viewpoint.

The theory of finite fields is a branch of modern algebra that has come to the fore in the last 50 years because of its diverse applications in combinatorics, coding theory, and the mathematical study of switching circuits, among others. The origins of the subject reach back into the 17th and 18th century, with such eminent mathematicians as Pierre de Fermat (1601–1665), Leonhard Euler (1707–1783), Joseph-Louis Lagrange (1736–1813), and Adrien-Marie Legendre (1752–1833) contributing to the structure theory of special finite fields—namely, the so-called finite prime fields. The general theory of finite fields may be said to begin with the work of Carl Friedrich Gauss (1777–1855) and Evariste Galois (1811–1832), but it only became of interest for applied mathematicians in recent decades with the emergence of discrete mathematics as a serious discipline.

In this book, which is the first one devoted entirely to finite fields, we have aimed at presenting both the classical and the applications-oriented aspect of the subject. Thus, in addition to what has to be considered the essential core of the theory, the reader will find results and techniques that are of importance mainly because of their use in applications. Because of the vastness of the subject, limitations had to be imposed on the choice of material. In trying to make the book as self-contained as possible, we have refrained from discussing results or methods that belong properly to algebraic geometry or to the theory of algebraic function fields.

Exponential sums are important tools in number theory for solving problems involving integers—and real numbers in general—that are often intractable by other means. Analogous sums can be considered in the framework of finite fields and turn out to be useful in studying the number of solutions of equations over finite fields (see Chapter 6) and in various applications of finite fields.

A basic role in setting up exponential sums for finite fields is played by special group homomorphisms called characters. It is necessary to distinguish between two types of characters—namely, additive and multiplicative characters—depending on whether reference is made to the additive or the multiplicative group of the finite field. Exponential sums are formed by using the values of one or more characters and possibly combining them with weights or with other function values. If we only sum the values of a single character, we speak of a character sum.

In Section 1 we lay the foundation by first discussing characters of finite abelian groups and then specializing to finite fields. Section 2 is devoted to Gaussian sums, which are arguably the most important types of exponential sums for finite fields as they govern the transition from the additive to the multiplicative structure and vice versa. They also appear in many other contexts in algebra and number theory. The closely related Jacobi sums are studied in the next section.

The theory of polynomials over finite fields is important for investigating the algebraic structure of finite fields as well as for many applications. Above all, irreducible polynomials—the prime elements of the polynomial ring over a finite field—are indispensable for constructing finite fields and computing with the elements of a finite field.

Section 1 introduces the notion of the order of a polynomial. An important fact is the connection between minimal polynomials of primitive elements (so-called primitive polynomials) and polynomials of the highest possible order for a given degree. Results about irreducible polynomials going beyond those discussed in the previous chapters are presented in Section 2. The next section is devoted to constructive aspects of irreducibility and deals also with the problem of calculating the minimal polynomial of an element in an extension field.

Certain special types of polynomials are discussed in the last two sections. Linearized polynomials are singled out by the property that all the exponents occurring in them are powers of the characteristic. The remarkable theory of these polynomials enables us, in particular, to give an alternative proof of the normal basis theorem. Binomials and trinomials—that is, two-term and three-term polynomials—form another class of polynomials for which special results of considerable interest can be established. We remark that another useful collection of polynomials—namely, that of cyclotomic polynomials—was already considered in Chapter 2, Section 4, and that some additional information on cyclotomic polynomials is contained in Section 2 of the present chapter.