basis for finite fields as a vector space

Question

If we consider GF(4) as a vector space over GF(2), the basis of GF(4) includes two elements 1 and a. Due to this fact that for an arbitrary vector space we can find several basis, what are other bases of GF(4) as a vector space over GF(2)?

Answer 1

We can write

$$\Bbb F_4=GF(4)=\{0,1,a,a+1\}\;,\;\;\text{with the condition}\;\;a^2=a+1$$

From the above representation it follows that $\;\{a\,,\,a+1\}\;$ is linearly independent over $\;\Bbb F_2\;$ and thus we have basis

$$\{1,a\}\;,\;\;\{1,a+1\}\;,\;\;\{a,a+1\}$$

and, of course, the same sets but in reverse order (as a basis is an ordered set of linearly independent vectors that generate the linear space)