University of Central FloridaSchool of Electrical Engineering and Computer ScienceEEL-6532: Information Theory and Coding.Spring 2010 - dcmHomework 3 - due Wednesday March 3, 2010Problem 1:Show that the necessary and sufficient condition forZnto be a finite field isthatnis prime. To do so prove that all properties of a finite field are satisfied whennis aprime number. To convince yourselves construct the addition and multiplication tables forZ6andZ7and identify the elements which do not have a multiplicative inverse whenn= 6.Problem 2:Show that there exists a characteristic element inGF(q); a characteristic elementgenerates all non-zero elements ofGF(q). To do so prove first that ifa∈GF(q), a6= 0 thenthe order ofadividesq-1.Problem 3:Show thataq=aif and only ifa∈GF(q).Problem 4:Show that ifαis a characteristic element ofGF(q) then∀a, b∈GF(q) we have(a+b)α=aα+bα.
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