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# chap7 - Chapter 7 Introduction to finite fields This...

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Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, partic- ularly groups, fields, and polynomials. Our primary interest is in finite fields, i.e., fields with a finite number of elements (also called Galois fields). In the next chapter, finite fields will be used to develop Reed-Solomon (RS) codes, the most useful class of algebraic codes. Groups and polynomials provide the requisite background to understand finite fields. A field is more than just a set of elements: it is a set of elements under two operations, called addition and multiplication, along with a set of properties governing these operations. The addition and multiplication operations also imply inverse operations called subtraction and division. The reader is presumably familiar with several examples of fields, such as the real field R , the complex field C , the field of rational numbers Q , and the binary field F 2 . 7.1 Summary In this section we brieﬂy summarize the results of this chapter. The main body of the chapter will be devoted to defining and explaining these concepts, and to proofs of these results. For each prime p and positive integer m 1, there exists a finite field F p m with p m elements, m and there exists no finite field with q elements if q is not a prime power. Any two fields with p elements are isomorphic. The integers modulo p form a prime field F p under mod- p addition and multiplication. The polynomials F p [ x ] over F p modulo an irreducible polynomial g ( x ) F p [ x ] of degree m form a finite field with p m elements under mod- g ( x ) addition and multiplication. For every prime p , there exists at least one irreducible polynomial g ( x ) F p [ x ] of each positive degree m 1, so all finite fields may be constructed in this way. Under addition, F p m is isomorphic to the vector space ( F p ) m . Under multiplication, the nonzero m 2 elements of F p m form a cyclic group { 1 , α, . . . , α p } generated by a primitive element α F p m . The elements of F p m are the p m roots of the polynomial x p m x F p [ x ]. The polynomial m x p x is the product of all monic irreducible polynomials g ( x ) F p [ x ] such that deg g ( x ) divides m . The roots of a monic irreducible polynomial g ( x ) F p [ x ] form a cyclotomic coset of deg g ( x ) elements of F p m which is closed under the operation of raising to the p th power. For every n that divides m , F p m contains a subfield with p n elements. 75

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76 CHAPTER 7. INTRODUCTION TO FINITE FIELDS For further reading on this beautiful subject, see [E. R. Berlekamp, Algebraic Coding The- ory , Aegean Press, 1984], [R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications , Cambridge University Press, 1986] or [R. J. McEliece, Finite Fields for Com- puter Scientists and Engineers , Kluwer, 1987], [M. R. Schroeder, Number Theory in Science and Communication , Springer, 1986], or indeed any book on finite fields or algebraic coding theory.
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