Define a field Using the addition and multiplication tables

Define a field. Using the addition and multiplication tables for the ring Z_3[x]/(d(x)) show that Z_3[x]/(d(x)) is a field. Explain your reasoning.

Solution

Field is a set with 2 binary operations defined on it, say + and * i.e., (F,+,*) such that:

1. (F,+) is abelian group and 2. (F,*) is abelian group.

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Here, it seems some information is missing! Could you please explain what is d(x). Since in the question it is mentioned that prove quotent is a field so definately d(x) will be a irreducible polynomial but what is it precisely!

Please mention in comment. I will get back to you with solution.