I got the assignment to determine the identification of $\mathbb{F}_3^2$ and $\mathbb{F}_{3^2}$. I am capable of constructing the non-prime fields $\mathbb{F}_{3^2}$ as a reduction of $\mathbb{F}_3$ by a irreducible polynomial of degree 3. By doing so I successfully obtain addition and multiplication tables.
What I do not understand is the connection between the two dimensional $\mathbb{F}_3^2$ and the one dimensional $\mathbb{F}_{3^2}$
- What does identification mean?
After having done some research I could work out the following: When constructing the finite field $\mathbb{F}_3^2$, we need to define addition and multiplication as well. While addition is trivial, multiplication seems to be a bit trickier. Therefore we somehow use the multiplication table of $\mathbb{F}_{3^2}$
- How does such "usage" look like?
- How can I multiply $\begin{pmatrix} 1 \\2 \end{pmatrix}$ by $\begin{pmatrix} 1 \\1 \end{pmatrix}$ making use of $\mathbb{F}_{3^2}$?
Can someone give me some good explanation that is suitable for a non-mathematician?