# Thread: Tables for a given field

1. ## Tables for a given field

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.

2. Originally Posted by page929
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 $\mathbb{Z}_3[x]/\langle x^2+x+2\rangle$. So the elements of the field are polynomials in x (with $\mathbb{Z}_3$ coefficients), but when you factor out the ideal generated by $x^2+x+2$ you are effectively equating $x^2+x+2$ with 0. Thus $x^2 = -x-2 = 2x+1$, and you can use that identity to eliminate all powers of $x$ higher than 1. Therefore the field contains nine elements:

$0,\ 1,\ 2,\ x,\ x+1,\ x+2,\ 2x, \ 2x+1,\ 2x+2.$

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 $2x+1$ for any occurrences of $x^2$.