Question

5. A field is a set F containing 0 and 1 that is an abelian group under addition, and (upon removing 0) Common examples of fields are abelian group under multiplication, for which the distributative law holds. an Q, R, and C. There is a unique finite field Fg of order q= p for every prime p and positive integer k. For all other q E N, there is no finite field of order g. For each of the fields F, F and F the Cayley diagrams for addition and multiplication are shown below, overlayed on the same set of nodes. The solid arTOws are the Cayley diagrams for addition and the dashed arrows are the Cayley diagrams for multiplication. e. (a) For each field above. determine whet her or not the addition and multiplication oper- ations are in fact, addition and multiplication modulo some number. If yes, relabel the vertices accordingly. If no, explain why it fails. (b) Create Cayley diagrams for the finite fields F andF

Answer 1

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