# hw3 - University of Central Florida School of Electrical...

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University of Central Florida School of Electrical Engineering and Computer Science EEL-6532: Information Theory and Coding. Spring 2010 - dcm Homework 3 - due Wednesday March 3, 2010 Problem 1: Show that the necessary and sufficient condition for Z n to be a finite field is that n is prime. To do so prove that all properties of a finite field are satisfied when n is a prime number. To convince yourselves construct the addition and multiplication tables for Z 6 and Z 7 and identify the elements which do not have a multiplicative inverse when n = 6. Problem 2: Show that there exists a characteristic element in GF ( q ); a characteristic element generates all non-zero elements of GF ( q ). To do so prove first that if a GF ( q ) , a 6 = 0 then the order of a divides q - 1. Problem 3: Show that a q = a if and only if a GF ( q ). Problem 4: Show that if α is a characteristic element of GF ( q ) then a, b GF ( q ) we have ( a + b ) α = a α + b α .
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