Let $F \leq E \leq K$ fields. Sea $\left\{\alpha_1, \alpha_2, \ldots \alpha_k\right\}$ a basis for the extension $K / E$ y $\left\{\beta_1, \beta_2, \ldots \beta_m\right\}$ a basis for the extension $E / F$. show that $\left\{\alpha_i \beta_j \mid 1 \leq i \leq k, 1 \leq j \leq m\right\}$ : has $k m$ distincts elements of $K$.
my try:
lets suppose that $\alpha_i\beta_j=\alpha_\ell\beta_n$ for $i \neq \ell$ and $j\neq n$
and then $\alpha_i\beta_j-\alpha_\ell\beta_n=0$ and how $\beta_j, \beta_n \subset E\subset K$ then are linearly independent in $K$ so $\alpha_i=\alpha_j=0$ which is a contradiction.