Minimal subfield of two fields

Question

Suppose $\Bbb F \subset \Bbb K_1 , \Bbb K_2 \subset \Omega$ all fields.

Denote by $\Bbb K_1\Bbb K_2$ the minimal field containing both $\Bbb K_1$ and $\Bbb K_2$

I need to prove that if $[\Bbb K_1:\Bbb F]$ and $[\Bbb K_2:\Bbb F]$ both finite then $[\Bbb K_1 \Bbb K_2 : \Bbb F]\le [\Bbb K_1:\Bbb F][\Bbb K_2:\Bbb F]$.

Not sure how to do it, hints/partial solution would be great.

Thanks !

Answer 1

Hint:

If $\;\{v_1,...,v_n\}\,,\,\,\{w_1,...,w_m\}\;$ are basis of $\;\Bbb K_1/\Bbb F\,,\,\,\Bbb K_2/\Bbb F\;$ resp., then

$$\;\Bbb K_1\Bbb K_2=\text{Span}_{\Bbb F}\,\{\,v_iw_j\}_{1\le i\le n,\,1\le j\le m}\;$$