lectures - MATH 523 LECTURE NOTES Summer 2008 These notes...

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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.

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lectures - MATH 523 LECTURE NOTES Summer 2008 These notes...

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