Question is to prove that :
For finite extensions $E/k$ and $F/k$ Prove that $[EF:k]\leq [E:k][F:k]$
where $EF$ is the smallest field extension which contains both $E$ and $F$
Supposing $E=k(x_1,x_2,\cdots,x_n)$ and $F=k(y_1,y_2,\cdots,y_m)$ we can see that $EF=k(x_1,x_2,\cdots,x_n,y_1,y_2,\cdots,y_m)$
So, $[EF:k]\leq m+n$ which is definitely less than $mn=[E:k][F:k]$
So, $[EF:k]\leq m+n\leq mn=[E:k][F:k]$
When we know a strong result that $[EF:k]\leq m+n$ why are we looking for a less strong result that $[EF:k]\leq m+n\leq mn=[E:k][F:k]$
This Question was asked by my friend and i was unable to convince him that this does not make sense....
To be frank, at this instant I do not see anything wrong here...
Please Help me to see that this is wrong...
In general the number of generators of a finite extension has nothing to do with the degree. Of course in your argument you could chose the $x_i$ and $y_i$ to be elements of a basis of the respective extension. But then the union of the two sets generates $EF$ as a field (or ring) over $k$, but not as a vector space. To get a set of generators as a vector space you must consider the products $x_iy_j$ too. Which leads to the bound $\leq mn$.