Rings and Fields - Write down the nine elements of F9

[cooljosh2k2]

Rings and Fields - Write down the nine elements of F9

Homework Statement

In F9 = Z/3Z, there is no solution of the equation x^2 = −1, just as in R. So “invent”
a solution, call it 'i'. Then 'i' is a new “number” which satisfies i^2 = −1. Consider
the set F9 consisting of all numbers a+bi, with a,b in F9. Add and multiply these
numbers as though they were polynomials in 'i', except whenever you get i^2 replace
it by −1.
(i) Write down the nine elements of F9 .
(ii) Show that every nonzero element of F9 has an inverse, so that F9 is a
field.

The Attempt at a Solution

I know I am supposed to show you that I've tried the question if i want an answer. Believe me, i have tried it. I am just really confused by the wording of the question and am not really sure what they are looking for in part a. Once i get part a, I am pretty sure id be able to get part b on my own.

 

[VeeEight]

You are told that F₉ = {a+bi | a,b in F₉}. The first part is asking you to try out all the elements. So for example, 1+2i is an element.
The second part is asking you to show that for all a+bi in F₉, there is some c+di such that (a+bi)(c+di) = 1. Try writing the inverse out using a and b in some way, then multiply it out to check if you are correct and get 1.