When is the Dimension of the Composite Field $[LM : K]$ Finite? I hate making homework threads, but this is something that's been bugging me all week.
We have $K \subseteq F$ are fields with $L, M$ intermediate fields.  The composite field $LM$ is defined as $L(M)$, or equivalently $M(L)$.\
When is the index $[L(M) : K]$ finite?  We have that $L(M)$ is the set of all evaluations $f(m_1, ... , m_t)/g(m_1, ... , m_t)$ where $m_1, ... , m_t$ are in $M$ and $f, g$ are polynomials in $L[X_1, ... , X_n]$.\
If $x_1, ... x_a$ is a basis for $M_K$ and $y_1, ... , y_b$ is a basis for $L_K$ then I think the elements $x_iy_j$ where $1 \leq i \leq a$ and $1 \leq j \leq b$ will span $LM$.  This is evident if $LM$ were defined to be the smallest integral domain containing $LM$ (that is, if $LM$ were equal to $L[M]$ and not $L(M)$) rather than the smallest field.  However, when we allow multiplicative inverses of linear combinations, the whole thing gets confusing and I'm unsure of how to represent an inverse of a linear combination of elements from $L$ and from $M$ as a linear combination of elements $x_iy_i$.
 A: If $L,M$ are algebraic over $K$, then also $LM$ is algebraic over $K$, since it is spanned by algebraic elements. It follows that we don't need fractions, i.e. that $L \otimes_K M \to LM$ is surjective. In particular, $[LM:K] \leq [L:K] \cdot [M:K]$. We see that if $L,M$ are finite over $K$, then also $LM$ is finite over $K$, and the converse is trivial. The proof also shows that

If $x_1, ... x_a$ is a basis for $M_K$ and $y_1, ... , y_b$ is a basis for $L_K$ then I think the elements $x_iy_j$ where $1 \leq i \leq a$ and $1 \leq j \leq b$ will span $LM$.

is correct.
A: Well, if both $L$ and $M$ are finite over $K$, certainly $LM$ (also written $L\vee M$) is also finite over $K$. Was that your question? The best way to approach this is to remember that when $\alpha$ is algebraic over $K$, $L\vee K(\alpha)=L(\alpha)$ and that $\bigl[L(\alpha)\colon L\bigr]\le\bigl[K(\alpha)\colon K\bigr]$. This, together with the fact that $M\supset K$ is a tower of simple extensions, should be enough to justify it all, though I haven’t stepped through all the points of the necessary argument and may have omitted something essential.
I do think that the approach of looking at bases doesn't help much in this case.
