My understanding is the following:
I think,here, we have only to verify this collection of linear combinations is a field.
For this, I don't know how to verify the existence of multiplicative inverses for the non zero elements in this collection, as the author only tells the closure of this collection under addition and multiplication.
Because It is obvious since K_1K_2=F(\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_m), these collection linear combinations is contained K_1K_2.
And one can see K_1 and K_2 are contained in that collection of linear combinations.
If we can prove that this collection of linear combinations is a field, then it is a field containing K_1 and K_2 and K_1K_2 is the smallest one. So we get K_1K_2 is contained in that linear combinations.
Hence we get K_1K_2 is equal to that collection of linear combinations and so the collection of \alpha_i\beta_j spans K_1K_2
Can anyone help me to get out of this doubt?
It holds if either $ K_1 $ or $ K_2 $ is algebraic extension.
Take $ x\neq0 $ of $ K_1K_2 $, $x = a_1b_1 +\cdots + a_nb_n\ (a_i\in K_1,\ b_i\in K_2) $.
If $ K_1 $ is algebraic extension, $x ^ {-1} \in K_2 (a_1,\cdots, a_n) = K_2 [a_1,\cdots, a_n] $.
So $x^{-1}\in K_1K_2$.