# 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. Foreachpr ime 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 ]o fdeg ree 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 feachpos it ivedegree 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 )e lementso f F p m which is closed under the operation of raising to the p th power. 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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## This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).

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

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