I am to construct the Cayley tables for the additive and multiplicative operation in $(\Bbb F_7, +, *)$. I have started by stating
The order (nr of elements) of a finite field must be a prime or prime power. $\Bbb F_7$ is a finite field with 7 elements, and any polynomial $g \in \Bbb F_7$ can be uniquely written as $g=a_0+a_1x+a_2x^2+fq$ for some $q \in \Bbb F_7$, and some $a_0,a_1,a_2 \in \Bbb F_7$
Previusly I have constructed Cayley tables for groups $G=(\Bbb Z_n, operation)$, is constructing the table for fields the same? If not, how do they differ? How can I construct this table when I know how to construct the table for f.ex. $(\Bbb Z_7, +)$ or $(\Bbb Z_7, *)$?
Edit:
I have understood that the tables for $(\Bbb F_7, +)$ is the same as the table for $(\Bbb Z_7, +)$, so now I'm wondering what the point of this exercise was. What am I supposed to learn from it? I had the impression I had to do some kind og plynomial division or something, but if the tables are the same I'm confused.
I have written the following:
The tables for $(\Bbb F_7, +, *)$ is the same as the table for $(\Bbb Z_7, +)$ $$ \begin{array}{c|clr} + & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 0 \\ 2 & 2 & 3 & 4 & 5 & 6 & 0 & 1 \\ 3 & 3 & 4 & 5 & 6 & 0 & 1 & 2 \\ 4 & 4 & 5 & 6 & 0 & 1 & 2 & 3 \\ 5 & 5 & 6 & 0 & 1 & 2 & 3 & 4 \\ 6 & 6 & 0 & 1 & 2 & 3 & 4 & 5 \\ \end{array} \quad \begin{array}{c|clr} * & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 0 & 2 & 4 & 6 & 1 & 3 & 5 \\ 3 & 0 & 3 & 6 & 2 & 5 & 1 & 4 \\ 4 & 0 & 4 & 1 & 5 & 2 & 6 & 3 \\ 5 & 0 & 5 & 3 & 1 & 6 & 4 & 2 \\ 6 & 0 & 6 & 5 & 4 & 3 & 2 & 1 \\ \end{array} $$
But how do I show that this differ from the table for $(\Bbb Z_7, +)$? Because I have just used the tables I already calculated in the previous exercise where i calculated the tables for $(\Bbb Z_7, +)$ and $(\Bbb Z_7, *)$. How can I mathematically show that these are table for $(\Bbb F_7, +)$ and $(\Bbb F_7, *)$? If I write something like "The tables are the same as for those of $(\Bbb Z_7, +)$ and $(\Bbb Z_7, *)$", I haven't really done anything in this exercise. I am missing something that shows that my solution is for fields. Or do you think I'm overthinking this? I don't think I am, it just seems too easy to just copy the tables.