I am working with a homework is about normal basis on fields GF and I want opinions and maybe if you can help me in some doubts.
1) Find normal basis of the field $GF(3^6)$ which is understood as a vector space over the filed $GF(3^2)$
Since the degree of $GF(3^6)$ over $GF(3^2)$ is a power of the characteristic, an element 'a'generates a normal basis iff its trace to $GF(3^2)$ is not zero, we can for instance take for 'a'= $0$ of the polynomail $x^3-x^2+1$
2) In the chosen normal basis find (previous exercise) find matrix that represents the operation of multiplication by a fixed element of this field.
I was thinking to convert my previous ecuation to $x^3=x^2-1$ and star find values ($x^4$, $x^5$, etc ) to $A = [a_1, a_2, ..., a_{n-1}]_{n1} $ but I don't know if I can convert it to a matrix.
Somebody has a opinion it is ok or not (1th exercise), or has a different solution, and maybe can help me with the second exercise.
Thank you!
Your solution to part 1) is ok. A zero $a$ of $x^3-x^2+1$ will, indeed, give you a normal basis $\mathcal{B}=\{a,a^9, a^{81}=a^3\}$.
I'm relatively sure that in part 2) you are asked to do the following. Let $$z=c_1a+c_2a^9+c_3a^{81}$$ be an arbitrary element of $GF(3^6)$ - written using the basis $\mathcal{B}$, so the coefficients $c_1,c_2,c_3\in GF(9)$ are the coordinates of $z$ w.r.t. $\mathcal{B}$. Consider the $GF(9)$-linear mapping $T:u\mapsto zu$. The question asks you to jot down the matrix of $T$ w.r.t. $\mathcal{B}$. To that end you need to write the elements $za^{9^i}$, $i=0,1,2$, in terms of the basis $\mathcal{B}$. Basically you need to figure out the products of the basis elements $a^{9^i}\cdot a^{9^j}$, and their coordinates w.r.t. $\mathcal{B}$.