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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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Maybe just a book on finite fields would be of use? There is one with precisely that name, by Rudolf Lidl and Harald Niederreiter. If you told us what you have already seen, or what you want to do, it'd be easier to help you. –  Mariano Suárez-Alvarez Oct 8 '10 at 21:00
    
I second the Lidl and Niederreiter reference but would like to add Michael Rosen's Number Theory in Function Fields as a possible reference. –  WWright Oct 8 '10 at 22:23

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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.

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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!

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Try the book Finite Fields and Applications.

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