I am looking for a good reference that could clearly explain finite field arithemtic. Specifically I want to understand the importance and utility of the field generator polynomial and why it is important in multiplication.
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One area where finite fields is widely applied is in the theory of error-correcting codes. A very nice book that treats these two subjects jointly is:
Error-Correcting Codes and Finite Fields by Oliver Pretzel; Clarendon Press, Oxford, 1992.
When I was an undergrad, I was trying to learn this material for the purpose of finite geometry. I found that even with little background in Abstract Algebra, that this book by Simeon Ball and Zsuzsa Weiner, was very readable. The beginning in particular focuses on what you asked about.
Not to mention, it's free!