If I understand correctly, the definition of the degree of a field extension $L/K$ is the dimension of $L$ over $K$ interpreted as a vector space. Now if the degree is $n < \infty$, the basis looks like $\{ 1, a_1, \dots , a_n\}$ where $a_i \in L - K$.
My question is:
Is it right to write $\{ 1, a, a^2, \dots , a^n\}$ for some $a \in L - K$ because of the primitive element theorem? If not, why exactly is the basis usually written $\{ 1, a, a^2, \dots , a^n\}$ instead of $\{ 1, a_1, \dots a_n\}$?