Give addition and multiplication tables for the field Z3/<x^2 + x + 2>.
I am unsure how to find the congruence classes.
Any help would be appreciated.
I think you mean . So the elements of the field are polynomials in x (with coefficients), but when you factor out the ideal generated by you are effectively equating with 0. Thus , and you can use that identity to eliminate all powers of higher than 1. Therefore the field contains nine elements:
To get the addition and multiplication tables, work out the sum and product of each pair of elements using ordinary mod 3 arithmetic, but then substitute for any occurrences of .