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Unformatted text preview: MATH 523 LECTURE NOTES Summer 2008 These notes are intended to provide additional background from abstract algebra that is necessary to provide a good context for the study of algebraic coding theory. In particular, we need to show how to construct all finite fields and we need to investigate their properties. 1 The definition of a field We begin with the definition of a field. In linear algebra, we need to be able to add, subtract, multiply, and divide scalars. Similarly, in working with polyno mials we need to be able to add, subtract, multiply, and divide the coefficients. This leads to the definition of a field, which generalizes the properties of the set Q of rational numbers, the set R of real numbers, and the set C of complex numbers. Let F be a set on which two binary operations are defined, called addition and multiplication, and denoted by + and respectively. Then F is called a field with respect to these operations if the following properties hold: (i) Closure. For all a,b F the sum a + b and the product a b are uniquely defined and belong to F . (ii) Associative laws. For all a,b,c F , a + ( b + c ) = ( a + b ) + c and a ( b c ) = ( a b ) c . (iii) Commutative laws. For all a,b F , a + b = b + a and a b = b a . (iv) Distributive laws. For all a,b,c F , a ( b + c ) = ( a b ) + ( a c ) and ( a + b ) c = ( a c ) + ( b c ) . (v) Existence of identity elements. The set F contains an additive identity element, denoted by 0, such that for all a F , a + 0 = a and 0 + a = a . The set F also contains a multiplicative identity element, denoted by 1 (and assumed to be different from 0) such that for all a F , a 1 = a and 1 a = a . (vi) Existence of inverse elements. For each a F , the equations a + x = 0 and x + a = 0 1 have a solution x F , called an additive inverse of a , and denoted by a . For each nonzero element a F , the equations a x = 1 and x a = 1 have a solution x F , called a multiplicative inverse of a , and denoted by a 1 . In addition to the examples Q , R , and C mentioned earlier, the set Z 2 of congruence classes of integers modulo 2 also satisfies the axioms of a field. This is an extremely important example because of our interest in coding theory. The addition and multiplication tables for Z 2 are given below. + [0] [1] [0] [0] [1] [1] [1] [0] [0] [1] [0] [0] [0] [1] [0] [1] If p is a prime number, then the set Z p of congruence classes modulo p forms a field, so there is no limit to the size of a finite field. We will show that it is also possible to construct fields with p n elements, for any prime p and any n 1. As an example, we next exhibit a field with 4 elements....
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This note was uploaded on 06/13/2011 for the course CRYPTO 101 taught by Professor Na during the Spring '11 term at Harding.
 Spring '11
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