Is \alpha in LK determined by polynomials and elements from subfields L and K?

[winter85]

Good day,

I just need someone to tell me if this is correct. If L and K are subfields of M, their composite LK is the smallest subfield of M that contains both L and K.

is this correct $$\alpha \in LK$$ if and only if there are positive integers n and m, polynomials $$f(x_1,x_2,...,x_n) \in L[x_1,...,x_n]$$ and $$g(x_1,x_2,...,x_m) \in L[x_1,...,x_m]$$, and elements $$a_1,...,a_n, b_1,...,b_m \in K$$ such that $$\alpha = \frac{f(a_1,...,a_n)}{g(b_1,...,b_m)}$$ ?

Thanks.

 

[fresh_42]

Looks right, since $\alpha$ is algebraic over $L$ and $K$, so any quotient of such numbers will do.