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This question already has an answer here:

I was asked specifically to construct a field of $81$ elements. Can someone explain how the process for this works? And how should I approach more questions like this?

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marked as duplicate by Gregory J. Puleo, Travis, Ilmari Karonen, Elliot G, Hans Lundmark Mar 2 at 19:03

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Hint: use an irreducible polynomial of degree $4$ over $GF(3)$

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