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Unformatted text preview: Lecture 17 6 Abstract algebra and coding theory 1 Groups 1 / 15 Introduction In coding theory we study the following question: Coding theory Suppose we want to transmit a block of message from a sender to a receiver, for example from a ground station to a satellite. It often happens that during transmission certain errors will be introduced into our message. In coding theory one tries to add certain check symbols to the message block to make it possible for the receiver to find out if errors have occured or even to correct some errors and determine to right message. To fully understand this theory one needs the concept of finite fields . Finite fields A field is by definition an algebraic object with an addition and multiplication that satisfy certain properties. Examples for fields are the real numbers R and the complex numbers C . A finite field has only finitely many elements. We will see that for any given power p r of a prime number p there exists a finite field whose number of elements is p r . To understand the construction of finite fields we first have to study groups and rings. 2 / 15 Groups I Let G be a set and G × G → G , ( a , b ) mapsto→ a · b a map. Definition The pair ( G , · ) is called a group if the following conditions are satisfied: 1 · is associative, i.e. a · ( b · c ) = ( a · b ) · c , for all a , b , c ∈ G . 2 There exists a neutral element e ∈ G such that e · a = a · e = a for all a ∈ G . 3 For every a ∈ G there exists an element b ∈ G such that a · b = b · a = e . The operation · is called group multiplication . The element b in the third property is called the inverse of a and is denoted by b = a − 1 . The neutral element is also sometimes denoted by 1. Note that this has in general nothing to do with the 1 as a natural number, it lives in a completely different set. Often the point in multiplication is omitted and we write a · b = ab . 3 / 15 Groups II A fundamental property of groups is the following. Theorem Every group has precisely one neutral element. Moreover, the inverse of any group element is uniquely determined....
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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.
 Winter '11
 PhanThuongCang

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